TWISTED DUALITY AND POLYNOMIALS OF EMBEDDED GRAPHS

JOANNA A. ELLIS-MONAGHAN∗ AND IAIN MOFFATT†

Abstract. We consider two operations on the edge of an embedded graph (or equivalently a ribbon graph):

giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations

give rise to an action of S3e(G), the ribbon group of G, on G. We show that this ribbon group action gives

a complete characterization of duality in that if G is any cellularly embedded graph with medial graph Gm,

then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic(as abstract graphs) to Gm. We then show how this group action leads to a deeper understanding of the

properties of, and relationships among, various graph polynomials such as the generalized transition poly-

nomial, an extension of the Penrose polynomial to embedded graphs, and the topological Tutte polynomialof Bollobas and Riordan, with implications for various knot and link invariants.

1. Introduction

There are many investigations into planar and abstract graphs that have not yet been fully extended intothe context of graphs embedded in closed surfaces (i.e. ribbon graphs). We explore two such extensions here.We broaden the notion of planar duality and its relation to medial graphs to twisted duality for embeddedgraphs. This allows us to give a complete characterization of all graphs with a given graph as their medialgraph. We also adapt various graph polynomials, such as the Penrose polynomial of [60] and the generalizedtransition polynomial of [24], to embedded graphs, as Bollobas and Riordan have done in extending the Tuttepolynomial to encode topological information of ribbon graphs in [9, 10]. Twisted duality then provides anespecially apt tool for determining not only properties of these polynomials, but also how they relate to oneanother.

Our investigation into extensions of duality and the medial graph begins with the following property ofplanar graphs. Any planar graph G may be drawn in the plane, and both its planar dual, G∗, and its medialgraph, Gm, constructed. Furthermore (G∗)m = Gm. In fact, any abstract graph G has a 2-cell embeddingin some surface, and from that embedding both a surface medial and a surface dual can be constructed,analogously to the planar case, and again the surface dual has the same medial graph as G. Now considerthat medial graph, H, as an abstract graph. What precisely is the set of embedded graphs each with amedial graph isomorphic (as an abstract graph) to H? To answer this question we need a more flexiblenotion of duality than simply reversing the roles of the faces and the vertices. We need the partial duality,or duality with respect to an edge, of [19], but we also need the operation of giving an edge a half-twist.The result of applying combinations of these two operations to an embedded graph G gives a twisted dualof G. These two operations give rise to an action of S3

e(G) on G, which we call the ribbon group action. Theribbon group action, in the context of embedded graphs, gives a full understanding of the relations amongembedded graphs and medial graphs in that if G is any cellularly embedded graph with medial graph Gm,then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic(as abstract graphs) to Gm.

Date: April 6, 2022.∗ Department of Mathematics, Saint Michael’s College, 1 Winooski Park, Colchester, VT 05439, USA.

jellis-monaghan@smcvt.edu† Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA;

The work of the first author was supported by the Vermont Genetics Network through Grant Number P20 RR16462 from the

INBRE Program of the National Center for Research Resources (NCRR), a component of the National Institutes of Health(NIH). This paper’s contents are solely the responsibility of the authors and do not necessarily represent the official views of

NCRR or NIH..

1

arX

iv:0

906.

5557

v1 [

mat

h.C

O]

30

Jun

2009

In another direction, most graph polynomials apply either to plane graphs (e.g. the Penrose polynomialof [60]) or to abstract graphs (e.g. the classical Tutte polynomial of [70, 71, 72]). A notable exception isthe extension of the Tutte polynomial to ribbon graphs by Bollobas and Riordan in [9, 10]. We continueour investigation into the extensions of properties of planar and abstract graphs by adapting the generalizedtransition polynomial of [24] to embedded graphs. We use the embedding of a graph locally at the vertices toidentify vertex state weights (weights for transition systems at the vertices) to achieve this. In this contextthe ribbon group action carries over to an action of the symmetric group on the vertex state weights, givinga twisted duality relation in terms of the weight systems for this polynomial.

The Penrose polynomial, defined implicitly for plane graphs by Penrose [60] in the context of tensordiagrams in physics, can be formulated in terms of vertex states of the medial graph. Thus, it can be givenas an evaluation of a generalized transition polynomial with appropriately chosen vertex state weights (seeJaeger [40] and [23] for the planar case). The generalized transition polynomial, which applies to embeddedgraphs, then naturally extends the Penrose polynomial to embedded, i.e. ribbon, graphs. We then usetwisted duality properties to strengthen and extend to embedded graphs many of the plane graph results forthe Penrose polynomial from [2] and [40]. For example, we find an expression for the Penrose polynomialof a plane graph as a sum of chromatic polynomials, where the sum is indexed by a subgroup of the ribbongroup. This generalizes a result of Aigner. A new equivalency statement for the Four Color Theorem followsfrom this.

With a different weight system, the generalized transition polynomial also agrees with the topochromaticpolynomial, a shift of the topological Tutte polynomial of Bollobas and Riordan of [9, 10], for both theoriginal and multivariate versions. This leads to twisted duality relations for the topochromatic polynomialas well. Thus, we conclude by applying the twisted duality results in this area as well. In fact, much of ourmotivation for this study came from recent results relating knots and embedded graphs, and particularly thevarious recent connections between various realizations of knot polynomials as embedded graph polynomials([17, 18, 19, 20, 27, 49, 55, 56, 57]). We give a fuller discussion of these interdisciplinary relations in Section 7.

Part of our interest in these problems derives from developing design strategies for self-assembling DNAnanostructures (see for example [43, 44, 62]). Because of the great promise of nanotechnology, especially forbiomolecular computing, but also, for example, drug delivery and biosensors (see [76, 48]), recent research hasfocused on DNA self-assembly of nanoscale geometric constructs, notably graphs. Several different graphshave been constructed from self-assembling DNA strands, including cubes [16], truncated octahedra [78], rigidoctahedra [63], and buckyballs [65]. An essential step in building a self-assembling DNA nano-construct,whether for biomolecular computing or physical applications, is designing the component molecules. Thetheory developed here for classifying graphs by their common medial graph suggests a possible efficientconstruction technique: first assemble a four-regular graph H, then use a biological process (such as anenzymatic action) to accomplish the splittings at the 4-armed branched junction molecules forming the 4-valent vertices of the graph. Double-stranded segments of DNA, with either an even or an odd number oftwists, might then be introduced, binding to sites at the split vertices. These would form the edge segmentsto complete any of the various twisted duals that have H as a medial graph. The vertices of this newconstruct are the cycles of H. This allows one four-regular graph to act as a single molecular “template”for constructing a whole class of nanostructures. The polynomials considered here, which are generallyintractable to compute with current technology, might, in principle, be found by a biomolecular computingprocess of physical manipulations (splittings, cutting, and merging) of the molecules.

2. Embedded graphs

We use the term “embedded graph” loosely to mean any of three equivalent representations of graphs insurfaces. We may think of an embedded graph as any of:

(a) a cellularly embedded graph, that is, a graph embedded in a surface such that every face is a 2-cell;(b) a ribbon graph, also known as a band decomposition;(c) an arrow presentation.

In subsequent sections we will use these equivalent representations interchangeably, using whichever bestfacilitates the discussion at hand. We assume the reader is familiar with cellular embeddings of graphs,but we briefly review ribbon graphs, arrow presentations, and the equivalences. We use standard notation:

2

* * * *

.

Figure 1. Equivalence of ribbon graphs and 2-cell embeddings.

V (G), E(G), and F (G) are the vertices, edges, and faces, respectively, of an embedded graph, while v(G),e(G), and f(G) are the numbers of such. We will say that a loop in an embedded graph is non-twisted if aneighbourhood of it is orientable, and we say that the loop is twisted otherwise.

Definition 1. Following [10], a ribbon graph G = (V (G), E(G)) is a (possibly non-orientable) surface withboundary represented as the union of two sets of topological discs: a set V (G) of vertices, and a set E(G) ofedges such that

i. the vertices and edges intersect in disjoint line segments;ii. each such line segment lies on the boundary of precisely one vertex and precisely one edge;iii. every edge contains exactly two such line segments.

Ribbon graphs are considered up to homeomorphisms of the surface that preserve the vertex-edge struc-ture. (In particular, the embedding of a ribbon graph in three-space is irrelevant.) We will say that a ribbongraph is plane if it is the neighbourhood of a plane graph, or equivalently, if the ribbon graph is a genus zerosurface. A ribbon graph is orientable if it is orientable as a surface. Ribbon graphs are band decompositions(see Gross and Tucker [33]), with the 2-band interiors removed (the 2-bands correspond to faces). Note thatin this context, f(G) is just the number of boundary components of the surface with boundary comprisingthe ribbon graph G.

Definition 2. From Chmutov [19], an arrow presentation consists of a set of circles, each with a collectionof disjoint, labelled arrows, called marking arrows, indicated along their perimeters. Each label appears onprecisely two arrows. Two arrow presentations are considered equivalent if one can be obtained from theother by reversing the direction of all of the marking arrows which belong to some subset of labels.

The equivalence of the three representations is given rigorously in, for example [33, 19]; however it isintuitively clear. If G is a cellularly embedded graph, a ribbon graph representation results from taking asmall neighbourhood of the embedded graph G. This can be thought of as cutting out the ribbon graphfrom the surface. Neighbourhoods of vertices of the graph G form the vertices of a ribbon graph, andneighbourhoods of the edges of the embedded graph form the edges of the ribbon graph.

On the other hand, if G is a ribbon graph, we simply sew discs into each boundary component of theribbon graph to get the desired surface.

A ribbon graph can be obtained from an arrow presentation by viewing each circle as the boundary of adisc that becomes a vertex of the ribbon graph. In this paragraph we will call these “vertex discs”. Edgesare then added to the vertex discs by taking a disc for each label of the marking arrows. In this paragraphwe will call these the “edge discs”. Orient the edge discs arbirtarily and choose two non-intersecting arcson the boundary of each of the edge discs. Orient these arcs according to the orientation of the edge disc.Finally, identify these two arcs with two marking arrows, both with the same label, aligning the direction ofeach arc consistently with the orientation of the marking arrow. See Figure 2.

Conversely, every ribbon graph gives rise to an arrow presentation. To describe a ribbon graph G asan arrow presentation, start by arbitrarily orienting and labelling each edge disc in E(G). This induces anorientation on the boundary of each edge in E(G). Now, on the arc where an edge disc intersects a vertexdisc, place a marked arrow on the vertex disc, labelling the arrow with the label of the edge it meets anddirecting it consistently with the orientation of the edge disc boundary. The boundaries of the vertex setmarked with these labelled arrows give the arrow marked circles of an arrow presentation. See Figure 3.

3

Two vertices and an edgedisc

aligning the arrows identifying arcs

Figure 2. Constructing a ribbon graph from an arrow presentation.

2

3

1 1

2

3

1

2

3

.

Figure 3. Equivalence of arrow presentations and ribbon graphs.

Figure 4. The formation of a medial ribbon graph.

Medial graphs play a central role throughout the rest of this paper. If G is cellularly embedded, weconstructed its medial graph Gm exactly as in the plane case, by placing a vertex of degree 4 on each edge,and then drawing the edges of the medial graph by following the face boundaries of G. There is a naturalembedding of the medial graph, viewed as a ribbon graph, into G viewed as a ribbon graph. This results fromdrawing the edges of Gm very close to the the edges of G in the surface, and taking a smaller neighbourhoodof Gm than the neighbourhood of G when cutting out the ribbon graphs from the surface. See Figure 4. Anexample of a medial ribbon graph is given in Figure 5.

Ribbon graphs provide a particularly apt model for our motivating application of DNA nanoconstructs.The techniques of [1, 78, 16], and in essence [63], use single strands of DNA to trace out each edge of agraph, once in each direction, binding to themselves along segments of Watson-Crick complements to formdouble-helixes along the edges and stable branched junctions at the vertices. The sequences of nucleotidesare carefully specified so that only the two sides of an edge are complements of, and hence bind to, eachother. A strand may thus be interpreted as a facial walk of a graph embedded in a surface (see [42] for thetopological formalism), or equivalently as the boundary of a ribbon graph, with orientability determined bythe parity of the number of twists in the double strands of DNA forming the edges.

3. Twisted duality and the ribbon group action

Duality in the plane is constrained by restricting the result of taking the dual to be again a plane graph.Working with embedded graphs enables greater flexibility. This has been realized with the generalized dualityof Chmutov [19], which involves dualizing with respect to individual edges, and hence moving out of theclass of plane graphs. Here we extend this idea further, allowing two operations on the edges of an embeddedgraph G: not only taking the dual with respect to an edge, but also giving a half-twist to an edge. Thesetwo operations give rise to a group action of Ge(G) on G, which we call the ribbon group action. The ribbongroup action is the foundation of many of the results in the later sections of this paper.

4

A ribbon graph G. The medial graph Gmdrawn inside G.

Gm presented as a ribbongraph.

Figure 5. An example of a medial ribbon graph.

τ

(ei ei

)= ei ei δ

(ei ei

)=

ei

ei

Figure 6. τ and δ with arrow presentations.

3.1. The ribbon group action. As often is the case, we begin with graphs equipped with a linear orderingon their edges, and then show that our constructions are independent of these orderings. We first give thetwist and dual operations with respect to single distinguished edges. We define the ribbon group action ofGe(G) on graphs with a linear ordering on their edges, where G ∼= S3. We then provide a more efficientnotation which is independent of the ordering of the edges. In the following subsection we also establishsome elementary properties of the ribbon group action.

We write G for the set of all embedded graphs considered up to homeomorphism (we will identify G withits homeomorphism class), and G(n) ⊆ G for those with n edges. Similarly, we write

Gor = {(G, `)|G ∈ G and l is a linear order of the edges}for the set of embedded graphs with ordered edges, and

Gor(n) ={

(G, `)|G ∈ G(n) and ` is a linear order of the edges}

for those with n edges.

Definition 3. Let (G, `) ∈ Gor and suppose ei is the ith edge in the ordering l. Also, suppose G is given interm of its arrow presentation, so ei is a label of a pair of arrows.

The half-twist of the ith edge is (τ, i)(G, `) = (H, `) where H is obtained from G by reversing the directionof exactly one of the ei-labelled arrows of the arrow presentation, as in Figure 6. Most importantly, Hinherits its edge order ` in the natural way from G.

The dual with respect to the ith edge is (δ, i)(G, `) = (H, `), where H is obtained from G as follows.Suppose A and B are the two arrows labelled ei in the arrow presentation of G. Draw a line segment withan arrow from the the head of A to the tail of B, and a line segment with an arrow from the head of B tothe tail of A. Label both of these arrows ei, and delete A and B with the arcs containing them. The linesegments with their arrows become arcs of a new circle in the arrow presentation of H. As with the twist,H here inherits its edge order l from G. See Figure 6.

We make the simple but important observation that these operations applied to different edges commute.

Proposition 4. If i 6= j and ξ, ζ ∈ {τ, δ}, then (ξ, i)((ζ, j)(G, `)) = (ζ, j)((ξ, i)(G, `)).

However, (τ, i) and (δ, i) do not commute when applied repeatedly to the same edge, and in fact we willsee they induce a group action of S3 on that edge.

We use the following notation to denote compositions applied to the same edge:

(ξζ, i)(G, `) := (ξ, i)((ζ, i)(G, `)),

where ξ, ζ ∈ {τ, δ}. We also define (1, i)(G, `) := (G, `). Thus, we can consider the action of (ξ, i) on (G, `)where ξ is a word in {τ, δ}.

5

Lemma 5. If (G, `) ∈ Gor then, for each fixed i,

(τ2, i)(G, `) = (δ2, i)(G, `) = ((τδ)3, i)(G, `) = 1(G, `).

Therefore, given a fixed i, there is an action of the symmetric group S3, with the presentation

S3∼= G := 〈δ, τ | δ2, τ2, (τδ)3〉,

on Gor.

Proof. The following calculations verify that (δ2, i)(G, `) = (τ2, i)(G, `) = ((τδ)3, i)(G, `) = 1(G, `). Since

(τ,i)7−→ (τ,i)7−→ ,

then (τ2, i) = 1. Also(δ,i)7−→ (δ,i)7−→ ,

giving the identity (δ2, i) = 1. Finally,

(τ,i)7−→ (δ,i)7−→ (τ,i)7−→

(δ,i)7−→ =(τ,i)7−→ (δ,i)7−→ ,

and thus ((τδ)3, i) = 1, completing the proof of the lemma. �

The action in Lemma 5 for fixed i now extends to a group action of S3e(G) on Gor(n).

Definition 6. We call S3n ∼= Gn the ribbon group for n edges and define the ribbon group action of the

ribbon group on Gor(n) by:

(ξ1, ξ2, ξ3, . . . , ξn)(G, `) = (ξn, n)((ξn−1, n− 1) . . . ((ξ2, 2)((ξ1, 1)(G, `))) . . .)= ((ξn, n) ◦ (ξn−1, n− 1) ◦ . . . ◦ (ξ2, 2) ◦ (ξ1, 1))(G, `),

where ξi ∈ G for all i.

With Definition 6, if (G, `) ∈ Gor(n), then we can view (τ, i) as an element of Gn of the form (1, . . . , τ, . . . , 1),with τ in the ith coordinate, and similarly for δ and the other elements of G.

3.2. Twisted duals. We now define twisted duality for graphs without any edge ordering. Our final defi-nition of a twisted dual of G will be of the form

GQξi(Ai),

where the Ai’s partition the edge set, and the ξi’s are in G. However, we need some preliminary definitionsto make sense of this expression.

We begin with an obvious proposition.

Proposition 7. If (G, `) ∈ Gor(n), ζ ∈ Gn, and σ ∈ Sn, then ζ(G, `) = σ(ζ)(G, σ(`)), where σ acts on ζ bypermuting the order of the elements of the n-tuple.

Proposition 7 assures that GQξ(A1) as given below is well-defined, that is, it is independent of the edge

ordering `.

Definition 8. Suppose G ∈ G(n), A,B ⊆ E(G) and ξ, ζ ∈ G. Define Gξ(A) as follows. Let ` be an arbitraryordering (e1, . . . , en) of the edges of G, and define ξA := (ε1, . . . , εn) ∈ Gn, where εi = ξ if ei ∈ A and εi = 1else. Then,

Gξ(A) := ξA(G, `).6

Moreover, we establish the following notational conventions:

Gξ(A)ζ(B) := (Gξ(A))ζ(B), and

Gξζ(A) := Gζ(A)ξ(A).

Proposition 9. If, for all i, we have ζi ∈ G and Bi ⊆ E(G), then any expression of the form

GQζi(Bi),

(where the product is not necessarily commutative) is equal to

GQ6i=1 ξi(Ai),

where the product is commutative, and where the Ai ⊆ E(G) are pairwise disjoint with ∪iAi = E(G), andwhere ξ1 = 1, ξ2 = τ, ξ3 = δ, ξ4 = τδ, ξ5 = δτ, ξ6 = τδτ ∈ G.

Proposition 9 follows from repeated applications of Definition 8, and the fact that the product commutesfollows from the fact that the Ai’s are disjoint and Proposition 4. Hereafter, we will customarily omit anyfactors of the form 1(A1) or ξi(∅) in these expressions. Also, if Ai is given explicitly by a list of edges, tosimplify notation, we will omit the set brackets, for example writing e.g. τ(e, f) for τ({e, f}).

As an example of the of Proposition 9, if G is an embedded graph with edges d, e, f, g, h, then

Gτ(d,e,f)δ(e,f,g) = (Gτ(d,e,f))δ(e,f,g) = Gτ(d)(τ(e,f)δ(e,f))(δ(g)) = Gτ(d)δτ(e,f)(δ(g)) = Gτ(d)δ(g)δτ(e,f).

The edge h is unaffected.Geometrically, these maps act on ribbon graphs in the following way: τ adds a half-twist to the edge e,

and δ forms the partial dual (introduced by Chmutov [19] and further studied in [57, 58]), at the edge e.Products of τ and δ are applied to the edge successively. An example of the actions of τ and δ on a ribbongraph is given in Example 10.

Example 10. If G is an embedded graph with E(G) = {e1, e2}, with the order (e1, e2), represented as aribbon graph and an arrow presentation as shown below,

G =

e1

e2

=

e1

e2

e1

e2

,

then we have

(τ, 1)(G) = Gτ(e1) =

e1

e2

e1

e2

=

e1

e2

and

(δ, 1)(G) = Gδ(e1) =e1

e2

e1

e2 = e1

e2

.

7

The full orbit of G is

e1

e2

e1

e2e1

e2e1

e2

e1

e2

e1

e2

e1

e2

e1

e2

(δ, 1)

(τ, 1) (1, τ)

(1, δ)

= =

(δ, 1)

(τ, 1)

(1, τ)

(1, δ)

(δ, 1)

(τ, 1)

(1, τ)

(1, δ)

We now have the following definition of a twisted dual of an embedded graph.

Definition 11. If G is an embedded graph, then H is a twisted dual of G if it can be written in the form

H = GQ6i=1 ξi(Ai),

where the Ai’s partition E(G), and the ξi’s are the six elements of G.Equivalently, if l is an arbitrary ordering of the edges of G, then H is a twisted dual of G if ξ(G, `) = (H, `)

for some ξ ∈ Ge(G), i.e. H is a twisted dual of G if (H, `) is in the orbit of (G, `) under the ribbon groupaction. Note that twisted duality is symmetric, so we may speak of H and G as being twisted duals of oneanother.

The linear ordering on the edges is necessary to define a group action (since there is no “universal edgeset” for graphs as there is, say, a universal point set for a class of matroids), but not necessary to thegeometric construction of twisted duals. Thus we will use Definition 6 when we wish to emphasize the groupaction, and Definition 11 to emphasize the geometry. However, since moving between the two viewpointsis so natural, our language may not always preserve the distinction, as for example, we may speak of Gn“acting” on an unordered graph.

One of our main interests in this paper is in the orbits Orb(G, `) := Ge(G)(G, `) = {ξ(G, `) | ξ ∈ Ge(G)}of the group action. With slight abuse of terminology, we define

Orb(G) := {H : (H, `) ∈ Orb(G, `) for some edge order `}.

3.3. Some notable subgroups of the ribbon group. Twisted duality generalizes the notion of the partialdual of a ribbon graph. Partial duality is intimately related with knot theory, and can be used to representlink diagrams ([20, 56, 18, 19]) without the need for using edge weights to record the under/over crossings ofthe link. We also note that some classic results on duality extend to partial and twisted duality, for examplesee [58], where the second author gave a characterization of partially dual graphs in terms of edge bijectionsbetween graphs in the spirit of Edmonds [21], and of course, the results presented later in this paper.

We take a moment now to place prior work on partial duality in the context of twisted duality.

Definition 12. Let Gpd = 〈δ | δ2〉 be the subgroup of G generated by δ. Then two twisted duals H and G

are said to be partial duals if (H, `) = ξ(G, `) for some ξ ∈ Ge(G)pd and some edge ordering `.

The following fact relating twisted duality and Euler-Poincare duality will be important later. Recall thatthe Euler-Poincare dual G∗ of an cellularly embedded graph G is constructed exactly as in the plane case byplacing a vertex in each face, and connecting two of these vertices with an edge whenever their faces sharean edge on their boundaries. In the context of ribbon graphs, G∗ is constructed by regarding the ribbongraph G as a punctured surface, filling in the punctures using a set of discs denoted V (G∗), then removingthe original vertex set V (G) (so G∗ = (V (G∗), E(G))).

Proposition 13 (Chmutov [19]). If G is an embedded graph, then

G∗ = Gδ(E(G)).8

A second subgroup of the ribbon group of interest here is the subgroup generated by twists: Gtw =〈τ | τ2〉 ≤ G. We will say that an embedded graph G is a twist of H if (G, `) = ξ(H, `) for some ξ ∈ G

e(G)tw

and some edge ordering `. We can use the orbit of the action of Gtw to describe a certain set of embeddingsof a combinatorial map.

A combinatorial map is a graph equipped with a cyclic order of the incident half-edges at each vertex. Weare interested in embeddings of a combinatorial map. Usually, one considers embeddings of a map into anorientable surface such that the images of the cyclic orders at the vertices agree with an orientation of thesurface. With this convention, the study of combinatorial maps is equivalent to the study of embeddings ofgraphs into orientable surfaces. Here, however, we would like to be able to embed combinatorial maps intonon-orientable surfaces, and to be able to embed combinatorial maps in such a way that the cyclic orders atthe vertices need not extend to an orientation of the surface. To do this we observe that a neighbourhoodof any point on a surface can be equipped with one of two local orientations. These local orientations neednot extend to an orientation of the surface. By a locally admissible embedding of a combinatorial map M ,we mean a cellular embedding of M into some surface such that the cyclic order at the vertices of M ispreserved with respect to one of the local orientation of a neighbourhood of the image of each vertex. Forexample, the 3-regular plane graph with two vertices and the 3-regular, two vertex graph embedded in thetorus are both locally admissible embeddings of the same combinatorial map, in addition, this map will havelocally admissible embeddings on non-orientable surfaces such as the real projective plane.

Proposition 14. Let M be a combinatorial map and G be any locally admissible embedding of M , then theset of twists of G is equal to the set of locally admissible embeddings of M , that is

{ξ(G, `) | ξ ∈ Ge(G)tw } = {Gτ(A) | A ⊆ E(G)} = {locally admissible embeddings of M},

where ` is any linear order of E(G).

Proof. The set of locally admissible embeddings of M can be constructed using ribbon graphs in the followingway. Label all of the edges of M . For each vertex v of M with labelled, cyclically ordered half-edges(ev1 , . . . , evk), construct a disc with labelled, cyclically ordered half-ribbons (ev1 , . . . , evk) incident to thatdisc. A ribbon graph can then be obtained by gluing the two ei labelled half-edges together for each i. Mis a deformation retract of any ribbon graph constructed this way, and thus the resulting ribbon graph isequivalent to an embedding of M . Moreover, it is clear that the set S of ribbon graphs constructed by gluingin this way is equivalent to the set of locally admissible embeddings of M . Thus it is enough to show thatthe set of ribbon graphs S can be constructed as the set of all twists of one of its elements. To do this, letG,H ∈ S. Then H can be obtained from G by twisting a number of the edges of G and by reversing thecyclic ordering of the half-ribbons around some vertices. To complete the proof, all we need to do is to showthat a ribbon graph obtained by reversing the cyclic ordering of the half-ribbons around a vertex can beobtained by twisting edges. This can be done by adding a half-twist to every edge incident to the vertex. �

3.4. Some properties of the ribbon group action. Having defined the ribbon group action of Gn onGor(n), it is natural to inquire what type of group action this is. We will see that in general, the groupaction is faithful, transitive, is not free and has no fixed points. However, it turns out that stronger, label-independent analogues of these properties hold for twisted duality. We will state and prove these strongeranalogous properties in Proposition 15 and then state and prove the ordered versions in the language ofgroup actions in Corollary 16.

Proposition 15.(i) Let ξ ∈ Gn, then ξ(G, `) ∈ {(G, π(`)) | π ∈ Sn} for all G ∈ Gn if and only if ξ = 1.

(ii) For all G ∈ Gn there is some ξ ∈ Gn such that ξ(G, `) /∈ {(G, π(`) | π ∈ Sn}.(iii) For all G ∈ Gn, there exists an G′ ∈ Gn such that G′ is not a twisted dual of G, if and only if n > 1

Proof. (i) The sufficiency is easily verified by calculation.To prove the necessity, for each ξ 6= 1 we find (G, `) ∈ Gor(n) with the property that, if ξ(G, `) =

(H, `), then G 6= H as embedded graphs, and hence (H, `) 6= (G, π(`)) for any ordering π(`).Given ξ 6= 1, so that ξ = (ξ1, ξ2, . . . , ξn) has ξi 6= 1 for some i, we consider (B, `), (B′, `) ∈

Gor(n), where B is the connected plane bouquet, which consists of n loops at a single vertex, and(τ, i)(B, `) = (B′, `).

9

If ξi = δ or τδ, then ξ(B, `) has more than one vertex (since (ξi, i)(B, `) has two vertices). Ifξi = τ then ξ(B, `) is non-orientable (as (ξi, i)(B, `) is non-orientable). If ξi = δτ or τδτ , thenξ(B′, `) has more than one vertex (since (ξi, i)(B′, `) has two vertices). An all of these cases the ξchanges the underlying graph as required.

(ii) Let (G, `) ∈ Gor(n), then G either contains a cycle or does not contain a cycle. If G contains a cyclethen there exists a set of edges A of G such that adding a half-twist to each of the edges in A willchange the orientability of G. Let ξ = (ξ1, . . . , ξn), where ξi = τ if ei ∈ A and 0 otherwise. It thenfollows that exactly one of (G, `) and ξ(G, `) is orientable.

On the other hand, if G does not contain a cycle, then taking the dual at any edge e of G will resultin a ribbon graph a ribbon graph containing a cycle. In this case, if we let ξ = ((δ, 1), (1, 2), . . . (1, n)),we have that exactly one of (G, `) and ξ(G, `) contains a cycle.

In either case there exists some ξ ∈ Gn with the property that the ribbon graphs (without anedge order) in (G, `) and ξ(G, `) are distinct, and therefore (G, `) 6= ξ(G, `1), for any edge order `1.

(iii) Sufficiency is easily checked by calculation. To prove the necessity, assume that n > 1. We need toshow that for any graph G ∈ Gn there is some graph H ∈ Gn which is not a twisted dual of G.

We will prove the result by showing that there exists a set S of ribbon graphs in Gn that is closedunder taking the twisted duals and that has the additional property that every orientable ribbongraph in the set is plane. Thus, any graph G ∈ Gn − S is not a twisted dual of any graph in S andvice versa. However, for all n > 1, there is an orientable non- plane ribbon graph, so Gn − S is notempty.

Our desired set S has the property that G ∈ S if and only if G is plane and has a distinguishedvertex v such that every edge in G is either a loop incident with v or the edge is a bridge incidentwith v and a 1-valent vertex; or G is a twist of such a ribbon graph. Notice that every orientableribbon graph in S is plane.

To show that S is closed under the operation of twisted duality it is enough to show that for eachG ∈ S, e ∈ E(G) and ξ ∈ G, we have Gξ(e) ∈ S. To see that this is indeed the case let e be an edge ofsome G ∈ S. Then e is either a non-twisted loop, a twisted loop or a bridge, and ξ = 1, τ, δ, τδ, δτ orτδτ . If e is a non-twisted loop then the edge corresponding to e in G1(e) and Gτδτ(e) is a non-twistedloop; in Gδ(e) and Gτδ(e) is a bridge; and in Gτ(e) and Gδτ(e) is a twisted loop. If e is a twisted loopthen the edge corresponding to e in Gτ(e) and Gτδ(e) is a non-twisted loop; in Gδτ(e) and Gτδτ(e) isa bridge; and in G1(e) and Gδ(e) is a twisted loop. If e is a bridge then the edge corresponding to ein Gδ(e) and Gδτ(e) is a non-twisted loop; in Gδτ(e) and Gτδτ(e) is a bridge; and in G1(e) and Gδ(e)

is a twisted loop. In all off these cases the resulting twisted dual Gξ(e) is in S and the result thenfollows.

�

Corollary 16. The action of Gn on Gor(n) is

(i) faithful if and only if n > 1;(ii) has no fixed points;

(iii) transitive if and only if n > 1;(iv) not free;

Proof. The first three properties follow easily from Proposition 15.To show that the action is not free we need to show that there is some (G, `) ∈ Gn, such that ξ(G, `) = (G, `)

for some non- trivial ξ ∈ Gn. This exists since∏ni=1(τ, i)(K1,n, `) = (K1,n, `), for any edge order ` of the

complete bipartite ribbon graph K1,n. �

Remark 17. In this paper we investigate and characterize the orbits of the ribbon group action. In additionto the orbits, the stabilizer subgroup of Ge(G), like the automorphism group, is an invariant of G, and thuswarrants further study. Another interesting question that we do not address here, is how ribbon graphtheoretic properties, such as the genus or number of vertices, vary over the elements in an orbit.

10

4. Medial graphs and the ribbon group action

Via the ribbon group action, twisted duality gives the full story of the interplay among a graph, its medialgraph, and its various twisted duals. This allows us to answer the questions we originally posed: Givenany 4-regular graph F , what precisely is the set of embedded graphs that have medial graphs isomorphicto F as abstract graphs? We are simultaneously able to provide a classification of all the twisted duals ofan embedded graph G via its medial graph, and thus characterize Orb(G). These relations among twistedduals, embedded medial graphs, and cycle family graphs are higher genus generalizations of the classic resultsrelating plane duals, medial graphs, and Tait graphs.

By way of motivation, we begin by reviewing some basic properties of plane medial graphs. If F isa connected 4-regular plane graph, then F is face two-colourable (see e.g. Fleischner [31]), and we callthis a checkerboard colouring, and use the colours black and white. The blackface graph, Fbl, of F is theplane graph constructed by placing one vertex in each black face and adding an edge between two of thesevertices whenever the corresponding regions meet at a vertex of F . The whiteface graph, Fwh, is constructedanalogously by placing the vertices in the white faces, and, borrowing terminology from knot theory, we referto Fbl and Fwh as the Tait graphs of F .

There are two key properties of Tait graphs. The first key property is duality: (Fbl)∗ = Fwh and vice versa,where the asterisk indicates planar duality. Thus, if G is any plane graph, and we give Gm the canonicalcheckerboard colouring, i.e. where the black faces contain the vertices of G, then

(1) (Gm)bl = G, and (Gm)wh = G∗.

Secondly, the medial graph of a Tait graph is just the original graph, i.e.

(2) (Fwh)m = (Fbl)m = F.

Moreover, {Fwh, Fbl} is exactly the set of plane graphs whose medial graph is F .With this, we can think of the Tait graphs loosely as “orbits” of size two under the operation of planar

duality, and everything in this “orbit” shares the same medial graph. In Section 4.3 we will see how thispoint of view is paralleled by embedded graphs.

In the special case that a 4-regular embedded graph F (thought of as being cellularly embedded in asurface) is checkerboard colourable, then we can construct the Tait graphs just as in the plane case, and thesame properties described above will still hold. In particular, there are the following two well-known results.

Proposition 18. If G is any embedded graph, thought of as being cellularly embedded in a surface, andwe canonically checkerboard colour the embedded medial graph Gm, then (Gm)bl = G and (Gm)wh = G∗ =Gδ(E(G)), the Euler-Poincare dual of G in the surface.

Proposition 19. Let F be a 4-regular, cellularly embedded graph. Then

(i) if F is checkerboard colourable, then {Fbl, Fwh} is the complete set of cellularly embedded graph withembedded medial graph equal to F ;

(ii) if F is not checkerboard colourable then F is not the embedded medial graph of any embedded graph.

The two propositions above provide a complete characterization of the embedded graphs whose medialgraph is equal to a given 4-regular, embedded graph F . Moreover, the propositions tell us that all of theembedded graphs with this property are duals.

A natural question then arises: which embedded graphs have a medial graph that is isomorphic to a given4-regular, embedded graph F? In this section we answer this question, giving a complete characterization ofthe set of embedded graphs with this property. To do this we introduce the concept of the cycle family graphsof a 4-regular, embedded graph F . The cycle family graphs do not rely on the checkerboard colourability ofan embedded graph and every 4-regular, embedded graph will admit a set of cycle family graphs. We willprove that for a given 4-regular, embedded graph F ,

(i) Gm ∼= F if and only if G is a cycle family graph of F (c.f. Proposition 19);(ii) Gm ∼= Hm if and only if G and H are twisted duals (c.f. Proposition 18).

Thus the relationships between cycle family graphs and twisted duals fully extend the classic relationsbetween the Tait graphs and duality.

11

4.1. Cycle family graphs. Let F be a 4-regular embedded graph thought of as a 2-cell embedding. Avertex state of v ∈ V (F ) is a choice of one of the following reconfigurations in a neighborhood of the vertexv:

, or .

The configurations replace a small neighbourhood of the vertex v. We will refer to the first two of the thesevertex states as splits and the third as a crossing. Vertex states are sometimes called transitions or transitionsystems, but here again we choose terminology that is closer to that of knot theory.

An arrow marked vertex state of v consists of a vertex state equipped with exactly two v-coloured arrows.Each arrow is placed on one of the positions indicated below and may point in either direction.

, , or .

A graph state s of F is a choice of vertex state at each vertex of F , and an arrow marked graph state ~sof F is a choice of arrow marked vertex state at each vertex of F . Note that each graph state correspondsto a specific family of disjoint cycles in F , and this family is independent of embedding (although differentembeddings of F will generally use different vertex states to generate the same family of disjoint cycles).

Definition 20. Let F be a 4-regular embedded graph, and let ~s be an arrow marked graph state of F .Regard ~s as an arrow presentation of an embedded graph by viewing each component of ~s as a circle markedby the labelled arrows arising from the arrow marked vertex states. Denote this new graph by F~s, andbecause the vertices of F~s arise from a family of disjoint cycles of F , we call F~s a cycle family graph of F .Also note that there is a natural identification between the vertex set of F the edge set of F~s. We denotethe set of all cycle family graphs of a 4-regular embedded graph F by C(F ).

Example 21. As an example, if F = * *v1

v2 , then one of the arrow marked graph state ~s of

F gives

F~s = * *v1 v2

v1v2 =

v1

v2

= * *v1

v2.

Example 22. As a second example of cycle family graphs, the reader can verify that the complete set of cycle

family graphs of contains only , and .

We will say that two arrow marked vertex states are equivalent if we can obtain one from the other byreversing the direction of pairs of arrows with the label, see Figure 7 for the case of splits. We refer to thearrowed states on the left of Figure 7, with the arrows pointing in opposite directions, as flat arrowed states,and the arrowed states on the right of Figure 7, with the arrows pointing in the same direction, as twistedarrowed states. Furthermore, we say that two arrow marked graph states ~s and ~s′ of F are equivalent if foreach vertex of F , the arrow marked vertex states of ~s and ~s′ at that vertex are equivalent when thought ofas arrow presentations.

12

∼ and ∼

Figure 7. Equivalent arrowed vertex states, with flat arrows on the left, and twisted arrowson the right.

Lemma 23. If F is a 4-regular embedded graph and ~s and ~s′ are two equivalent arrow marked graph statesof F , then F~s = F~s′ as embedded graphs.

Proof. We need to show that the arising arrow presentations, F~s and F~s′ are equivalent, that is, F~s′ canbe obtained from F~s by reversing the direction of some of the pairs of arrows of the same colour and byhomeomorphism of the cycles. To do this, it is enough to show that the arrow marked graph state ~s′ canbe obtained from ~s by reversing the direction of some of the pairs of arrows of the same colour and byhomeomorphism of the cycles. The fact that this is indeed the case is easily verified by checking the definingrelations of equivalent arrow marked graph and vertex states. �

We note that the converse of the above lemma is false as non-equivalent arrow marked graph states mayresult in equivalent cycle family graphs.

Corollary 24. There are at most 6v(F ) distinct cycle family graphs of a 4-regular ribbon graph F .

At certain points in this paper we will be particularly interested duality states. These are arrowed stateswhich arise by restricting all of the vertex states to splits with flat arrows. In particular, Tait graphs arisefrom special duality states, as follows.

Proposition 25. Let F be a checkerboard coloured embedded 4-regular graph. Then there exist arrow markedgraph states ~b and ~w of F such that F~b = Fbl and F~w = Fwh. Moreover, both ~b and ~w are duality states.

Proof. Construct an arrow marked graph state of F by choosing the duality state consisting of flat splitswhere the split at each vertex results in arcs that follow the boundaries of the black regions at that vertex.The resulting ribbon graph F~b is exactly the Tait graph Fbl since the cycles follow the boundaries of theblack regions (giving one vertex in each black region), and, since the arrow marked state is flat, there is anedge added whenever two black regions meet at a vertex as prescribed by the definition of Fbl. The whitefaceresult is proved analogously by choosing the flat splits at each vertex that follow the boundaries of the whiteregions. �

4.2. Twisted duals and cycle family graphs. We are now ready for the first of our main theorems. Wewill begin by showing that all of the cycle family graphs of a 4-regular embedded graph are twisted duals,thus generalizing the well known property of Equation 1. We will then prove the converse of this result, thattwisted duals are exactly the cycle family graphs of a medial graph. This converse generalizes Proposition 18.

Theorem 26. If F is a 4-regular cellularly embedded graph and F~s and F~s′ are two cycle family ribbongraphs, then Fs and Fs′ are twisted duals.

Proof. It is enough to show that if the arrow marked graph states ~s and ~s′ differ at exactly one vertex thenF~s and F~s′ are twisted duals. To show this, assume that ~s and ~s′ differ at the vertex v ∈ V (F ) and that ev isthe label of the edge in F~s and the corresponding edge in F~s′ that arises from the pair of v-coloured arrowsin ~s and ~s′. We then need to show that ξ(F~s, ev) = (F~s′ , ev) for some ξ ∈ G. To show this, we may assumewith out loss of generality (as G is a group) that the arrow marked vertex state at v in ~s is a split with flat

arrow markings. Locally, (the arrow presentation of) F~s is then . Then, in ~s′ the arrow presentation

of F~s′ is locally one of

= τ

, = δ

, = τδ

,

13

= δτ

or = τδτ

,

where the arrow presentations are all identical outside of the region shown in the diagrams. Thus we haveshown that when ~s and ~s′ differ at v, then ξ(F~s, ev) = (F~s′ , ev) for some ξ ∈ G, as required. �

As mentioned above, the converse of Theorem 26 holds: all partial duals are cycle family graphs of a4-regular embedded graph. Theorems 26 and 27 give a characterization of the orbits of the ribbon groupaction: Orb(G) = C(Gm). This means that the ribbon group action on an embedded graph G generatesprecisely the set of cycle family graphs of medial graph of G.

Theorem 27. If G is an embedded graph, then the set of cycle family graphs of Gm is precisely the set ofall twisted duals of G, i.e.

C(Gm) = Orb(G).

Proof. By Theorem 26, if H ∈ C(Gm), then H ∈ Orb((Gm)~s)) for any arrow marked state ~s. In particularthis is true for the arrow marked state ~b corresponding to the black face graph that is guaranteed byProposition 25, so that (Gm)~b = (Gm)bl = G. Thus C(Gm) ⊆ Orb(G).

To show that C(Gm) ⊇ Orb(G), we need to show that if G and G′ are twisted duals of one another, thenthey are both cycle family ribbon graphs of the medial ribbon graph Gm of G. To show this, consider locally

an edge e of G: . At the edge e, the twisted dual will take on one of the following six forms.

, , , , , or .

To prove the result it is enough to show that each of these six forms arise as an arrow marked vertex state ofve ∈ V (Gm). (This is since the local configurations at the edges of the arrow presentation of G are connectedto each other in the same way that the local configurations at the vertices of Gm are connected to each otherto form the cycles.)

Now the vertex ve of Gm is embedded in the edge e of G thus: . We then see that the

possible arrow marked vertex states of Gm at ve are

, , ,

, , and ,

as required. Thus, if H ∈ Orb(G), then H is a twisted dual of G, and by the above H ∈ C(Gm) andOrb(G) ⊆ C(Gm). Thus Orb(G) = C(Gm) as desired. �

We note that Theorem 26 does not follow from Theorem 27 since, unlike the planar case, there are 4-regular embedded graphs that do not arise as the medial graph of an embedded graph. (To see this, observethat there are three embedded graphs with exactly one edge, giving rise to three medial graphs with twoedges. However, as there are more than three 4-regular embedded graphs with two edges, not every 4-regularembedded graph can be an embedded medial graph.)

It follows immediately from Proposition 25 that Theorem 27 generalizes Equation 1 and Propositions 18,since (Gm)bl = (Gm)~b and (Gm)wh = (Gm)~w are both in C(Gm), and G and G∗ = Gδ(E(G)) are both inOrb(G).

We have seen the relation between cycle family graphs and twisted duals. Since partial duality is a specialcase of twisted duality that is of independent interest, we now specialize our results to partial duality.

14

Theorem 28. (i) If F is a 4-regular ribbon graph and F~s and F~s′ are cycle family ribbon graphs, forsome ~s and ~s′ that are duality states, then F~s and F~s′ are partial duals.

(ii) If G and G′ are partial duals, and Gm is the embedded medial graph of G, then there are dualitystates ~s and ~s′ such that G = (Gm)~s and G′ = (Gm)~s′ .

(iii) (Gm)~s is a partial dual of G if and only if ~s is a duality state.

Proof. The first two parts of the theorem can be proved by following the proof of Theorem 27 and restrictingto partial duality and duality states. Due to this similarity the proofs are omitted.

The proof of (iii) is as follows. Since Gm is a 4-regular ribbon graph and G = (Gm)~s for some dualitystate ~s, necessity follows by (i). For sufficiency, we may assume without loss of generality that (Gm)~s isobtained from G by forming the partial dual at a single edge e (so (Gm)s = Gδ(e)). If e is the edge ,

then the corresponding edge in (Gm)~s is , where the graphs are identical except in the region shown.

It is easily seen that the only way that this configuration can arise from an arrow marked vertex state atve ∈ V (Gm) is if the arrowed vertex state is a flat split. �

4.3. Medial graphs and cycle family graphs. In this subsection we will prove another of our maintheorems, this one generalizing Equation 2 and Proposition 19. The theorem shows that cycle family graphsextend the essential relations among plane graphs and their medial and Tait graphs. Furthermore, we showthat if F is a 4-regular graph, then its set of cycle family graphs is precisely the set of embedded graphs thathave embedded medial graphs isomorphic to F as abstract graphs. We actually prove a stronger result: wenot only show that the set of cycle family graphs of F give all the embedded graphs whose medial graphs areisomorphic to F as abstract graphs, but we also give specific conditions for when a medial graph is simplya twist of F , that is, a different embedding of the same combinatorial map as in Subsection 3.3.

Theorem 29. If F is a 4-regular embedded graph and ~s is an arrow marked graph state of F , then:(i) if ~s is a duality state, (F~s)m and F are twists of one another, i.e. (F~s)m = F τ(A) for some A ⊆ E(F );

(ii) otherwise (F~s)m and F are isomorphic as abstract graphs.

Proof. In order to prove the statements it is enough to consider what happens locally at a vertex v of F inthe formation of (F~s)m. We assume that v is incident with edges labelled av, bv, cv, dv in the cyclic order(av bv cv dv)

To prove the first statement, assume that ~s is a duality state. Without loss of generality we may assumethat the cycles defining the cycle family graph F~s travel between edges av and bv and between cv and dv.This is shown as the first step in the figure below.

av

bvcv

dv av

bvcv

dv av

bvcv

dv

Now taking the medial graph of F~s will add a vertex incident with edges labelled av, bv, cv, dv with the cyclicorder (av bv cv dv) as shown in the figure above possibly with additional twisting of the edges. Note thatsince we do not know if the arcs (av, bv) and (cv, dv) are connected to each other, we do not know if the edgeev of the cycle family graph will embed in the vertex v shown the first figure. This means that when weform the medial graph (Fs)m, as shown in the second step of the figure, we know nothing about the twistingof the edges. We acknowledge that the figure is slightly misleading in this respect.

Finally, since up to twisting of the edges, F and (F~s)m are identical at the vertex v ∈ F and thecorresponding vertex in (F~s)m for each vertex v of F and the endpoints marked av, bv, cv, dv in the figureare connected in the same way in F and in (F~s)m, it follows that these two embedded graphs, when viewedas arrow presentations, have the same sequence of labels on the vertex disc, although possibly not the samedirections of the arrows. Thus, (F~s)m = F τ(A) for some A ⊆ E(G).

This completes the proof of the first statement.15

The second statement is proven similarly: all that is needed to adapt the proof is to check that theremaining vertex states at v lead to an appropriate configuration at the corresponding vertex in (F~s)m. Thisis seen by the following calculations:

av

bvcv

dvav

bv cv

dv

,

av

bv cv

dvav

bv cv

dv

,

av

bv cv

dvav

bv cv

dv

.

There is a natural identification between the vertex set of F and the edge set of F~s, and a naturalidentification between the edges set of F~s and the vertex set of (F~s)m, and hence a natural identificationbetween the vertices of F and the vertices of (F~s)m. Thus, the second statement follows from observing thatthese local configurations mean that a pair of vertices are connected by an edge in F if and only if they areconnected by an edges in (F~s)m. �

Again by Proposition 25, it is clear that Theorem 29 generalizes Equation 2 and Proposition 19, sinceFbl = F~b and Fwh = F~w for the duality states ~b and ~w.

With the following theorem we are now able to answer the original question of finding the exact set ofembedded graphs that have medial graphs isomorphic to a given 4-regular graph F .

Theorem 30. If F is any 4-regular embedded graph, then the set of cycle family graphs of F is precisely theset of all embedded graphs G such that Gm ∼= F as abstracts graphs, i.e.

C(F ) = {G|Gm ∼= F}.

Proof. If G ∈ C(F ), then G = F~s for some arrowed state ~s, and hence Gm = (F~s)m ∼= F by Theorem 29. Onthe other hand, if G is an embedded graph with with Gm ∼= F , then Gm is checkerboard colourable with theblack faces containing the vertices of G. We take the arrowed state ~b of Gm guaranteed by Proposition 25so that (Gm)~b = (Gm)bl = G. We use the isomorphism between Gm and F to transfer the cycles andarrow markings to the original embedding of F to get an arrow marked state ~s′. Then G is equal to F~s′ asembedded graph since they have the same arrow presentations. Thus G ∈ C(F ) as desired.

�

Theorem 30 provides a second characterization of the orbit of an embedded graph under the ribbon groupaction:

Corollary 31. If G is an embedded graph and ~s is an arrow marked state of the embedded medial graph Gm,then ((Gm)~s)m ∼= Gm, i.e.

Orb(G) = {H : Hm∼= Gm}.

Proof. This follows immediately from Theorems 27 and 30.�

As an example of these results, the orbit of the plane digon in Example 10 gives the complete set of ribbongraphs with medial graphs isomorphic to the graph consisting of two vertices with four edges in parallel.The duality states give rise to this graph in the plane (for G and G∗) and on the torus (for the partial duals

16

with respect to just one of the edges). Note that flipping one vertex of the torus embedding gives a twist(on all the edges) of the planar embedding, thus illustrating part (i) of Theorem 29.

4.4. Cycle family graphs and checkerboard colourings. Here we define an action of the ribbon groupG on an arrow marked vertex state of an embedded medial graph Gm. Obviously, Gm could equivalently beany 4-regular checkerboard colourable embedded graph, but we will typically apply these results to medialgraphs. In order to do this we need to be able to distinguish among the three vertex states. If we canonicallycheckerboard colour Gm, then we can distinguish among the vertex statesat v as in the following figure.Here, following knot theory conventions, the graphs are identical outside these local neighborhoods.

v−→

in Gm white split black split crossing.We denote the graphs that result from each of these vertex states as follows:

(Gm)wh(v) is the embedded graph that results from taking the white split state at v,(Gm)bl(v) is the embedded graph that results from taking the black split state at v,and (Gm)cr(v) is the embedded graph that results from taking the crossing state at v.

We now define an action of the group G on an arrow marked vertex state. Let (G, `) ∈ Gor(n). Then theorder ` of the edges of G induces an order ˆ = (v1, . . . , vn) of the vertex set of the embedded medial graphGm. We let (Gm, ˆ) denote this canonically checkerboard coloured medial graph equipped with the inducedvertex order from (G, `). To define an action on the set of arrow marked states of Gm we let ~s be an arrowmarked graph state of Gm and (~s, i) be the vertex state at the ith vertex vi in the order ˆ. Then defineτ(~s, i) to be the pair (~s′, i), where ~s′ is obtained from ~s by reversing exactly one of the arrows of the arrowmarked vertex state at vi in ~s.

Let δ(~s, i) be the pair (~s′, i) where ~s′ is constructed by changing the arrow marked vertex state at vi in ~sas specified by the following figure:

δ

δ

δ

δ

δ

δ

.

Next we want to extend this action to the set of states of Gm. Let Cor(Gm, ˆ) denote the set of arrowmarked graph states of (Gm, ˆ) with the vertex ordering induced by `. Define an action of Gn on Cor(Gm, ˆ)by

ξ(~s, ˆ) := (ξ1, ξ2, ξ3, . . . , ξn)(~s, ˆ) = ξn(· · · ξ3(ξ2(ξ1(~s, v1), v2), v3) · · · ), vn),

where (ξ1, ξ2, ξ3, . . . , ξn) ∈ Gn and ~s ∈ Cor(Gm, ˆ).

Proposition 32. The action of Gn on Cor(Gm, `) described above is a group action. Moreover,

(Gm, ˆ)ξ(~s) = ξ((Gm, ˆ)~s),

where ξ ∈ Gn.

Proof. To prove the first part of the proposition we need to check that δ2(~s, v) = τ2(~s, v) = (τδ)3(~s, v) = 1and this is easily verified. The second part of the proposition is tautological when the arrowed states areviewed as arrow presentations of embedded graphs.

�

Proposition 32 may be given from a geometric perspective as follows.17

Proposition 33. Let G be an embedded graph with embedded medial graph Gm, and index the vertices ofGm by E(G). Let Γ =

∏6i=1 ξi(Ai) where the Ai’s partition E(G), and the ξi’s are the six elements of G. If

~s is an arrow marked state of Gm, then let ~sΓ be the result of applying ξi(~s, ve) if e ∈ Ai. Then

(Gm)~sΓ = ((Gm)~s)Γ.

Remark 34. Returning for a moment to the motivating application in self-assembling DNA nanostructures,we note that if a 4-regular graph F can be assembled out of DNA strands, with relatively small faces, then itmight be used as a template to construct all of the graphs in C(F ). The branched junction molecules formingits vertices would be decomposed into various vertex states, giving a state ~s of F . This kind of split at thevertices was done to find Hamilton circuits by Adelman [1]. Then long (relative to the faces of F ) doublestrands of DNA with ‘sticky ends’ (i.e. an extended single strand of unsatisfied bases) might be introducedto then form the edges of F~s, with the relatively small cycles of ~s as the vertices.

4.5. Deletion, contraction, and the medial graph. We now discuss how the operations of deletion andcontraction interact with forming medial graphs. Deletion and contraction of non-loop edges of an embeddedgraph are defined much as for abstract graphs. Let G be an embedded graph. In the language of ribbongraphs it is clear that deleting any edge of G or contracting a non-loop edge will result in a ribbon graph.When working in the language of cellularly embedded graphs, one has to ensure that deleting or contractingan edge results in a cellularly embedded graph. In particular, deleting a bridge changes not only the numberof components of a graph, but the number of components of the surface it is embedded in. Thus deletion andcontraction is often best done by converting to the language of ribbon graphs, carrying out the operation,then converting back to the language of cellularly embedded graphs. Deletion and contraction for arrowpresentations can be defined similarly. Note that G/e and Gδ(e) − e are equal.

However, contracting loops requires some care. We follow Bollobas and Riordan’s definition from Section 7of [10]. Let G be an embedded graph regarded as a ribbon graph and suppose e is a loop of G, with v thevertex of G incident to e. Then the ribbon subgraph ({v}, {e}) of G has either one or two boundarycomponents (depending on whether e is a twisted loop or not). Then the ribbon graph G/e is formed fromG by attaching a vertex-disc to each of the boundary components of v ∪ e in G, then deleting e and v. Fromthis definition, it is not hard to see that when e is a loop it is also true that the ribbon graphs G/e andGδ(e) − e are equal. We use this observation to define the contraction of a loop e of any embedded graph Gby setting

G/e := Gδ(e) − e.The following proposition may be readily observed by viewing G and Gm as ribbon graphs as in Figure 4,

noting how the medial graph is transformed under the indicated operations.

Proposition 35. Let G be an embedded graph with embedded medial graph Gm, and e be an edge of G.(i) (Gm)bl(ve) = (G− e)m.

(ii) If e is not a non-twisted loop, then (Gm)wh(ve) = (G/e)m.(iii) If e is not a twisted loop, then (Gm)cr(ve) and (Gτ(e)/e)m are twists of each other.

5. The transition polynomial

The generalized transition polynomial, q(G;W, t), of [24] is a multivariate graph polynomial that gen-eralizes Jaeger’s transition polynomial [40]. The transition polynomial assimilates the Penrose polynomialand Kauffman bracket, and agrees with the Tutte polynomial via a medial graph construction. We willnow adapt q(G;W, t) to embedded graphs and determine its interaction with the ribbon group action. Thiswill allow us, in Section 6, to generalize the Penrose polynomial for plane graphs to embedded graphs anddetermine new properties for it. Also, in Section 7, by leveraging the relation determined in [27, 30] betweenthe generalized transition polynomial and the topological Tutte polynomial of Bollobas and Riordan ([9, 10]),we determine new properties of the latter.

5.1. The topological transition polynomial. The generalized transition polynomial, q(G;W, t) extendsthe transition polynomial of Jaeger [39] to arbitrary Eulerian graphs and incorporates pair and vertex stateweights. For the current application, however, we will restrict q to 4-regular embedded graphs (typicallymedial graphs) and we will only work in the generality needed for our current application. For example,

18

since we will not use pair weights here, we give the weight systems simply in terms of vertex state weights. Ifan applications arises in the future where the pair weights are needed, because of the restriction to 4-regulargraphs, they can be taken to be square roots of the vertex state weights. We refer the reader to [24] or [30]for further details.

A weight system, W (F ), of any 4-regular graph F (embedded or not) is an assignment of a weight in aunitary ring R to every vertex state of F . (We simply write W for W (F ) when the graph is clear fromcontext.) If s is a state of F , then the state weight of s is ω(s) :=

∏v∈V (F ) ω(v, s), where ω(v, s) is the vertex

state weight of the vertex state at v in the graphs state s. Note that a state s consists of a set of disjointclosed curves, and we refer to these as the components of the state, denoting the number of them by c(s).

Definition 36. Let F be a 4-regular graph having weight system W with values in a unitary ring R. Thenthe state model formulation of the generalized transition polynomial is

q(F ;W, t) =∑s

ω(s)tc(s),

where the sum is over all graph states s of F .

We now restrict our attention further to embedded medial graphs and particular weight systems deter-mined by the embeddings. Because of these restrictions, we will call the generalized transition polynomialspecialized for this application the topological transition polynomial, and define it as follows.

Definition 37. Let G be an embedded graph with embedded medial graph Gm. Define the medial weightsystem Wm(Gm) using the canonical checkerboard colouring of Gm. A vertex v has state weights given byan ordered triple (αv, βv, γv), indicating the weights of the white split, black split, and crossing state, inthat order. We write (α,β,γ) for the set of these ordered triples, indexed equivalently either by the verticesof Gm or by the edges of G. Then the topological transition polynomial of G is:

Q(G, (α,β,γ), t) := q(Gm;Wm, t).

Proposition 38. (see [27]) The topological transition polynomial may also be computed by repeatedly applyingthe following linear recursion relation at each v ∈ V (Gm), and, when there are no more vertices of degree 4to apply it to, evaluating each of the resulting closed curves to t:

q(Gm,Wm, t) = αvq((Gm)wh(v),Wm, t) + βvq((Gm)bl(v),Wm, t) + γvq((Gm)cr(v),Wm, t).

Pictorially, this is:

v= αv + βv + γv .

Example 39. For example, if G = u v , then Gm =u v

and so

Q(Gm; (α,β,γ), t) = αuv

+ βuv

+ γuv

= αu

(αv + βv + γv

)+ · · ·

= αuαvt+ αuβvt2 + αuγvt+ · · ·

Theorems 42 and 41 give properties of the topological transition polynomial that will allow us to easily ma-nipulate related polynomials such as the topological Penrose polynomial of Section 6 and the topochromaticpolynomial of Section 7. Theorem 41 shows precisely how the topological transition polynomial interactswith the ribbon group action, and Theorem 42 is a modified deletion/contraction reduction for well-behavededges.

19

5.2. Twisted duals and the transition polynomial. The group G = 〈δ, τ | δ2, τ2, (τδ)3〉 is isomorphicto S3 via

η : τ 7→ (1 3) and η : δ 7→ (1 2).

Furthermore, the symmetric group S3 acts on the ordered triple of the weight system at a vertex by permu-tation. This action by S3 on a vertex can be extended to an action of Sn3 on medial graphs with n linearlyordered vertices. This can be done by mimicking the approaches used in Subsections 3.1 and 4.4. We willnot formally define this action here and instead define the order independent analogue of the action, whichis more convenient for our applications. This allows us to use the ribbon group to modify the medial weightsystem of an embedded medial graph.

Definition 40. Let Gm be a canonically coloured embedded medial graph of G with medial weight systemWm (or equivalently (α,β,γ)), and vertices indexed by the edges of G. Let Γ =

∏6i=1 ξi(Ai) where the

Ai’s partition E(G), and the ξi’s are the six elements of G. Then WmΓ (or (α,β,γ)Γ), the weight system

permuted by Γ, has the ordered triple of the weight system at a vertex ve given by η(ξi)(αve , βve , γve) whene ∈ Ai.

The classical Tutte polynomial has the duality property that the Tutte polynomial of a plane graph G isthe same as that of its planar dual G∗ with the roles of the variables x and y permuted. In [27, 30] it wasshown that the topological transition polynomial has the duality property that

(3) q(Gm;Wm, t) = q(G∗m;W ∗m, t),

or equivalently,

(4) Q(G; (α,β,γ), t) = Q(G∗; (β,α,γ), t),

where G∗ is the Euler-Poincare dual of G and W ∗m is the weight system that derives from exchanging theorder of αv and βv in the weight system at each vertex (i.e. take Γ = δ(E(G)) in Definition 40). Thisled to a new duality result for the topological Tutte polynomial of Bollobas and Riordan, also in [27, 30].We are now able to extend this Euler-Poincare duality to a full twisted duality property for the topologicaltransition polynomial. This twisted duality relation says that the topological transition polynomial of themedial graph of G is the same as that of the medial graph of any of the twisted duals, provided the weightsystem is appropriately permuted. We will apply this twisted duality relation in the subsequent sections toderive new duality properties for the Penrose and topological Tutte polynomials.

Theorem 41. Let G be an embedded graph with embedded medial graph Gm, and let Γ =∏6i=1 ξi(Ai) where

the Ai’s partition E(G), and the ξi’s are the six elements of G. Then,

q (Gm;Wm, t) = q(GΓm;WΓ

m, t),

or equivalently,

Q(G; (α,β,γ), t) = Q(GΓ, (α,β,γ)Γ, t).

Proof. It suffices to prove this for one edge at a time, i.e. to prove the property for Gξ(e) where ξ ∈ (G). Theeffect of ξ(e) on the medial graph is shown in the table below. The first row gives ξ. The second row showsthe effect on the arrow presentation of the arrows labelled e in the disc presentation of Gξ(e). The labels∗, a, b, c are the labels on the disc(s) on either side of e, and the configurations of the discs are otherwise

identical (so that, for example, if∗

b a

c

represents two discs, then∗ b

ac

represents one disc, and vice

20

versa). The third row shows the changes in the cyclic order of the vertices about ve in (Gξ(e))m.

ξ : 1 τ δ τδ δτ τδτ

Gξ(e) :

∗

b a

c ∗

b

a

c

∗ b

ac

∗

b

a

c

∗ b

a c

∗

ba

c

(Gξ(e))m :

∗

B A

C ∗

B

A

C

∗ B

AC

∗

B

A

C

∗ B

A C

∗

BA

C

Note that if we give the labels a, b, c the order (a, b, c), then they are permuted in the second row of the tableaccording to η(ξ), and hence (αve , βve , γve) are permuted as claimed. We illustrate this for Gτδ(e) below,leaving the other cases to the reader.

Gm =

∗

B A

C ∗

B A

C ∗∗

B A

C

BA

C

(Gτδ(e))m =

∗

B

A

C

∗

B

A

C

∗∗

B

A

C B

A

C

�

Note that by taking take Γ = δ(E(G)), the result of Equation 3 is now just an immediate corollary ofTheorem 41.

We use the notation (0,β,γ) to denote a weight system (α,β,γ) with the property that αi = 0 for all i.We use a similar notation when the αi, βi or γi are equal to 0 or some constant for each i.

Theorem 42. Let G be an embedded graph and e ∈ E(G).(i) If e is not a loop of G, then

Q(G; (α,β,γ), t) = αeQ(G/e; (α,β,γ), t) + βeQ(G− e; (α,β,γ), t) + γeQ(Gτ(e)/e; (α,β,γ), t).

(ii) If e is a non-twisted loop of G, then

Q(G; (0,β,γ), t) = βeQ(G− e; (0,β,γ), t) + γeQ(Gτ(e)/e; (0,β,γ), t).

(iii) If e is a twisted loop of G, then

Q(G; (α,β,0), t) = αeQ(G/e; (α,β,0), t) + βeQ(G− e; (α,β,0), t).

Proof. These identities follow from Propositions 35 and 38. �

Corollary 43. Let G be an embedded graph, e ∈ E(G) and ξ ∈ G. Further, let (α,β,γ)ξ(e) = (α′,β′,γ′).(i) If e is not a loop of Gξ(e), then

Q(G; (α,β,γ), t) = α′eQ(Gξ(e)/e; (α′,β′,γ′), t)+β′eQ(Gξ(e)−e; (α′,β′,γ′), t)+γ′eQ(Gτξ(e)/e; (α′,β′,γ′), t).

(ii) If e is a non-twisted loop of Gξ(e) and (α,β,γ)ξ(e) = (0,β′,γ), then

Q(G; (α,β,γ), t) = β′eQ(Gξ(e) − e; (0,β′,γ′), t) + γ′eQ(Gτξ(e)/e; (0,β′,γ′), t).

(iii) If e is a twisted loop of Gξ(e) and (α,β,γ)ξ(e) = (α′,β′,0), then

Q(G; (α,β,γ), t) = α′eQ(Gξ(e)/e; (α′,β′,0), t) + β′eQ(Gξ(e) − e; (α′,β′,0), t).21

Proof. These identities follow easily from Theorems 42 and 41. �

6. The Penrose polynomial

6.1. The Penrose polynomial for embedded graphs. We apply the results of Section 5 to the Penrosepolynomial, P (G,λ). This polynomial graph invariant for plane graphs was originally defined implicitlyby Penrose [60] in the context of tensor diagrams in physics, but turned out to have remarkable graphtheoretic properties. Excellent graph theoretical expositions are given by Aigner in [2, 3, 4], with additionalexploration of its properties by Aigner and Mielke [5], Ellis-Monaghan and Sarmiento [23], Sarmiento [61],and Szegedy [66]. In this section we use the generalized transition polynomial to define a “topological”Penrose polynomial that extends the original the Penrose polynomial to embedded graphs, much as Bollobasand Riordan have extended the Tutte polynomial to embedded graphs in [9, 10].

Given its origins, the Penrose polynomial has some surprising properties, particularly with respect tograph colouring. The Four Color Theorem is equivalent to showing that every plane, cubic, connected graphcan be properly edge-coloured with three colours. The Penrose polynomial, when applied to plane, cubic,connected graphs, encodes exactly this information (see Penrose [60]): if G is a plane, cubic, connectedgraph, then

P (G; 3) =(−14

) v(G)2

P (G;−2) = the number of edge 3-colourings of G.

One of our goals of this section is to determine properties of the Penrose polynomial that can be extendedto embedded graphs. We give counter examples for some of those that can not, and new results, includingcolouring results, for a number that can. We also use the ribbon group action to find new identities for thePenrose polynomial on embedded graphs. (We will not continue to use the adjective ‘topological’, insteadreferring to both the original polynomial and the extension defined here simply as the Penrose polynomial.)

Like the generalized transition polynomial, of which it is a specialization, the original Penrose polynomialof a graph G can be computed using either a state model formulation or a linear recursion relations appliedto its medial graph (see Jaeger [40] for example). We will use both approaches here.

Definition 44 (The original Penrose polynomial of a plane graph). Let G be a plane graph with medialgraph Gm, let St(Gm) be the set of states of Gm, and let St′(Gm) be the set of states with no black splits.Let WP be the medial weight system with αv = 1, βv = 0, and γv = −1 for all v ∈ V (Gm). Then,

P (G;x) =∑

s∈St(Gm)

ωP (s)xc(s) =∑

s∈St′(Gm)

((−1)cr(s) xc(s)

),

where c(s) is the number of components in the graph state s, and cr(s) is the number of crossing vertexstates in the state s.

Example 45. As an example, if G is the plane θ-graph consisting of two vertices joined by three edges inparallel, then P (G;x) = x3 − 3x2 + 2x, as shown below.

22

-x +x -x2 +x

+x -x2 +x3 -x2

G = Gm =

Computing the Penrose polynomial of a graph $G$ from the states of its medial graph.

The Penrose polynomial may also be computed via a linear recursion relation (see Jaeger [40] for example),by repeated applying the skein relation

= − .

to any vertex of degree 4 in Gm, and at the end, evaluating each of the resulting cycles to x.This immediately suggests how the Penrose polynomial can be extended to embedded graphs via the topo-

logical transition polynomial. Definition 46 generalizes the relation between the original Penrose polynomialand the transition polynomial given by Jaeger [40] and Ellis-Monaghan and Sarmiento [24, 23], where inthe latter properties of the generalized transition polynomial are used to derive enumeration formulae forthe original Penrose polynomials. Here we will similarly use properties determined in the Section 5 for thetopological transition polynomial to find new identities for the Penrose polynomial on embedded graphs.

Definition 46. Let G be an embedded graph and Gm be its canonically checkerboard coloured embeddedmedial graph. The Penrose weight system WP (Gm) is defined by letting αv = 1, βv = 0, and γv = −1 forall v ∈ V (Gm) in the medial weight system. We will also denote this weight system by (1,0,−1).

The Penrose polynomial P (G;λ) ∈ Z[λ] of an embedded graph G is then

P (G;λ) = q(Gm;WP , λ) = Q(G; (1,0,−1), λ),

where Q is the topological transition polynomial.

We will define the set of Penrose states, P(G), of a ribbon graph G to be the set of all graph states suchthat each vertex state is either a white split or a crossing. The Penrose polynomial can then be expressedas the state sum

(5) P (G;λ) =∑

s∈P(G)

(−1)cr(s)λc(s).

We can use this state sum to express the Penrose polynomial of a ribbon graph G in terms of twistedduals of G, as opposed to an expression in terms of its medial graph Gm.

23

Proposition 47. Let G be a ribbon graph, then

P (G;λ) =∑

A⊆E(G)

(−1)|A|λf(Gτ(A)).

Proof. There is a one-to-one correspondence between the subsets A ⊆ E(G) and flat arrow marked Penrosestates of Gm by taking a flat crossing ve in ~sA whenever e ∈ A, and taking a flat white split else. Clearlycr(sA) = |A|. If ~b is the state of Gm consisting of all flat black splits, so G = (Gm)~b, then the flat arrowmarked Penrose state corresponding to A ⊆ E(G) can be written as ~bδ(E−A)δτ(A). We then have

(Gτ(A))∗ = (((Gm)~b)τ(A))∗ = (Gm)~bδ(E−A)δτ(A) ,

where the second equality follows from Proposition 33.Note that f F is a 4-regular embedded graph, s is a state of F , and ~s any arrow marking of s, then

f((F~s)∗) = c(~s) = c(s), since f((F~s)∗) = v(F~s) = c(~s) = c(s).Thus, since Gτ(A) = ((Gτ(A))∗)∗,

f(Gτ(A)) = f(((Gτ(A))∗)∗) = f(((Gm)~bδ(E−A)δτ(A))∗) = c(~bδ(E−A)δτ(A)) = c(sA).

The result then follows.�

This expression of the Penrose polynomial in terms of the set of twists of a ribbon graph will be convenientlater.

6.2. Twisted duality and identities for the Penrose polynomial. One of the major advantages ofconsidering the topological Penrose polynomial is that it satisfies various identities that are not realizable interms of plane graphs. Many of these identities arise by considering twisted duality. For example, we see inthis context that the Penrose polynomial has deletion-contraction reductions similar to those of the Tuttepolynomial.

Proposition 48. The Penrose polynomial of an embedded graph G has the following properties.(i) If A ⊂ E(G) then P (G;λ) = (−1)|A|P (Gτ(A);λ), and in particular |P (G;λ)| is an invariant of the

orbits of the twist. Furthermore, P (G;λ) = (−1)|A|Q(G,WPτ(A), λ).

(ii) If e ∈ E(G), thenP (G;λ) = P (Gδ(e);λ)− P (Gδτ(e);λ).

(iii) If e is a non-twisted loop of G that bounds 2-cell, then P (G;λ) = (λ− 1)P (G− e;λ).(iv) The Penrose polynomial satisfies the four-term relation:

P

;λ

− P ;λ

= P

;λ

− P ;λ

,

where the four ribbon graphs in the figure are identical except in the region shown.

Proof. We will prove the properties one by one.(i) Recall that W τ(A)(Gm) is the weight system defined by

W τ(A)(Gm) :ve

=

− if e /∈ A

− if e ∈ A

.

Thus,P (Gτ(A);λ)

= Q(Gτ(A); (1,0,−1), λ)

= Q((Gτ(A))τ(A); (1,0,−1)τ(A), λ)24

= Q(G; (1,0,−1)τ(A), λ)

P (Gτ(A);λ)

= (−1)|A|Q(G; (1,0,−1), λ)

,where the second equality follows from Theorem 41 and the fourth equality is because for each

Penrose state s of Gm we have that (−1)|A|ωP (s) = ωPτ(A)(s). This proves both parts of Item (i).

(ii) We begin with the equation

(6) q(Gm;WP , λ) = q(Gm;W δ(e)P , λ)− q(Gm;W τδ(e)

P , λ).

To see why this equation holds, we note that by applying Proposition 38 to the weight systems WP ,W

δ(e)p and W

τδ(e)p respectively we have

q(Gm;Wp, λ) = q((Gm)wh(ve);Wp, λ)− q((Gm)cr(ve);Wp, λ),

q(Gm;W δ(e)p , λ) = q((Gm)bl(ve);W

δ(e)p , λ)− q((Gm)cr(ve);W

δ(e)p , λ)

and

q(Gm;W τδ(e)p , λ) = −q(Gm(Gm)wh(ve);W

τδ(e)p , λ) + q((Gm)bl(ve);W

τδ(e)p , λ).

Substituting the three identities above into the left and right hand sides of Equation 6 will give therequired equality.

We will now express each of the terms in Equation 6 in terms of the Penrose polynomial. Bydefinition, we have

(7) P (G;λ) = q(Gm;WP , λ).

For the second term we have

P (Gδ(e);λ) = q((Gδ(e))m;WP , λ)

= q(((Gδ(e))δ(e))m;W δ(e)P , λ)

= q(Gm;W δ(e)P (Gm), λ),

(8)

where the first equality follows by definition, the second follows from Theorem 41, and the thirdfollows from the fact that (δ(e))(δ(e)) = 1(e).

Finally, we can rewrite third term of Equation 6 as follows.

P (Gδτ(e);λ) = q((Gδτ(e))m;WP , λ)

= q(((Gδτ(e))τδ(e))m;W τδ(e)P , λ)

= q(Gm;W τδ(e)P , λ),

(9)

where the first equality follows by definition, the second follows from Theorem 41, and the thirdfollows from the fact that (δτ(e))(τδ(e)) = 1(e).

The result stated in the proposition then follows by substituting the identities in Equations 7, 8and 9 into Equation 6.

(iii) The proof of this property is straight forward and is therefore omitted.(iv) The fact that the state sum defining the Penrose polynomial satisfies the four-term relation is a well

known fact in the theory of Vassiliev invariants of knots. This is since the Penrose polynomial canbe expressed in term of the soN weight system. This point will be discussed in a little more detailin Remark 50 below. Since this fact is well known to knot theorists, we will only sketch a proof ofthe final property here.

Let Gi, i = 1, . . . , 4 be the four embedded graphs shown in the four-term relation in the ordershown in the defining figure. There is a canonical bijection between their edge sets. Identify the

25

edge sets of each of the Gi using this correspondence. Let e and f denote the distinguished edges ofthe Gi. Then since

P (Gi;λ) =∑

A⊆E(Gi)−{e,f}B⊆{e,f}

(−1)|A∪B|λf(Gτ(A∪B)i ),

it is enough to show that for a fixed subset A of E(Gi)− {e, f}, we have∑B⊆{e,f}

(−1)|B|λf(Gτ(A∪B)1 ) −

∑B⊆{e,f}

(−1)|B|λf(Gτ(A∪B)2 ) −

∑B⊆{e,f}

(−1)|B|λf(Gτ(A∪B)3 )

+∑

B⊆{e,f}

(−1)|B|λf(Gτ(A∪B)4 ) = 0.

This identity is easily verified by calculation.�

The following theorem provides deletion/contraction reductions for the Penrose polynomial, like those forthe Tutte polynomial, albeit (if we may be allowed one pun) with a slight twist.

Theorem 49. Suppose G is an embedded graph, and e ∈ E(G).(i) If e is not a loop of G, then

P (G;λ) = P (G/e;λ)− P (Gτ(e)/e;λ).

(ii) If e is not a loop of Gδ(e), then

P (G;λ) = P (Gδ(e) − e;λ)− P (Gτδ(e)/e;λ).

(iii) if e is a loop of neither G nor Gδ(e), then

P (Gδτ(e);λ) = P (G/e;λ)− P (G− e;λ),

or equivalently,P (G;λ) = P (Gτδ(e)/e;λ)− P (Gτδ(e) − e;λ).

Proof. Items (i) and (ii) follow immediately from Theorem 42 and Corollary 43. For Item (iii), apply Item(i) to Gδ(e), then add and subtract P (Gδ(e) − e;λ) and use Item (ii) to get that P (G;λ) − P (Gδ(e);λ) =P (G− e;λ)− P (G/e;λ). The result then follows from Proposition 48 Item 2.

�

The identities from Proposition 48 (together with the multiplicity under disjoint union) can be usedto reduce the calculation of the Penrose polynomial of an arbitrary embedded graph to its calculation onconnected, orientable bouquets. We would like to be able to reduce the calculation of the polynomialon these bouquets to its calculation on a manageable set of bouquets (as one can do with Bollobas andRiordan’s topological Tutte polynomial [9, 10]). This might enable one to provide a “universality property”or “recipe theorem” for the Penrose polynomial along the lines of the universality properties for the Tuttepolynomial [12, 13] or for Bollobas and Riordan’s topological Tutte polynomial [9, 10]). The fact that thePenrose polynomial satisfies the four-term relation might prove to be useful in this direction. For example,by adapting the proof of the main theorem in [15], one can reduce the evaluation of Penrose polynomial onblossoms, to the valuation of the Penrose polynomial of spine diagrams (these are chord diagrams with onechord that intersects all of the other chords).

Remark 50. The Penrose polynomial appears in the theory of Vassiliev invariants of knots as the the weightsystem ϕsoN (G) associated with the Lie algebra soN and its standard N -dimensional representation definedin [6, 7]. (See also [14] for an excellent introduction to Vassiliev invariants and the combinatorics of weightsystems.) Specifically, we have that for connected bouquets, 2e(G)P (G;N) = ϕsoN (G) where the ribbongraph G is regarded as a chord diagram on the right-hand side. (See [9, 10] for the relation betweenbouquets and chord diagrams.) The fact observed in Proposition 48 that the Penrose polynomial satisfiesthe four-term relation follows immediately.

26

A particularly provoking observation is that on connected bouquets, not only is the Penrose polynomialthe soN weight system, but Bollobas and Riordan’s topological Tutte polynomial [9, 10] is essentially theslN weight system. It remains a fundamental problem to fully understand the connection between these twoembedded graph polynomials, the slN and soN weight systems from the theory of Vassiliev invariants, andthe Homfly [34, 59] and Kauffman [46] polynomial (these are the knot invariants corresponding to the twoweight systems).

6.3. The Penrose polynomial and colourings. The Penrose polynomial of a plane graph is known tosatisfy several combinatorial identities and has numerous connections with graph colouring. It is naturalto ask which of these properties hold off of the plane. Here we will discuss relations between the Penrosepolynomial for non-plane graphs and graph colouring. Before moving on to the discussion of graph colouring,we observe that many of the basic properties of the Penrose polynomial of a plane graph given by Penrosein [60] and Aigner in [2] do not hold for non-plane graphs.

For example, the follow properties proved by Aigner in [2] do not hold for embedded graphs in general.

(i) If G is plane and Eulerian then P (G; 2) = 2v(G). This does not hold in general (consider ).(ii) If G is plane and has two regions with a common boundary e, then P (G;λ) = 2P (G/e;λ). This is

not true for general embedded graphs (consider ).(iii) If G is plane then the leading term of P (G;λ) is 1. This is not true for general embedded graphs

(consider ).(iv) If G is plane then the degree of P (G;λ) is the number of faces of G. This is not true for general

embedded graphs (consider ).

(v) If G is plane and Eulerian then P (G;−1) = 2e(G). This does not hold in general (consider ).

(Although |P (G;−1)| < 2e(G) for all G.)In [2], Aigner proved the following theorem which relates the number of proper k colourings of a graph

and the Penrose polynomial.

Theorem 51. Let G be a plane graph, then for all k ∈ N we have

χ(G∗; k) ≤ P (G; k).

In Theorem 54, we will complete Theorem 51 by showing that the Penrose polynomial of plane graph Gis in fact equal to a sum of specific chromatic polynomials. Moreover, this sum is indexed by the orbit of thetwist-action of G. It turns out that the expression χ(G∗; k) is a single summand in our expression for thePenrose polynomial P (G; k). Theorem 51 then follows from Theorem 54 as a corollary.

We will need the concept of an admissible k-valuation.

Definition 52. Let G = (V (G), E(G)) be an embedded graph and Gm be its medial ribbon graph. Ak-valuation of Gm is a k-edge colouring φ : E(GM )→ {1, 2, . . . , k} such that for every vertex ve of Gm, thenumber of i-coloured incident edges is even for each colour i.

A k-valuation is said to be admissible if at each vertex of Gm the k-valuation is of one of the followingtwo types:

i

j

i

j

i

ji

j

,

where i 6= j. The two local configurations above correspond to white split states and crossing states respec-tively.

The following theorem, which expresses the Penrose polynomial of a plane graph in terms of k-valuations,is due to Jaeger ([39] Proposition 13, see also [2] Proposition 4).

Theorem 53. If G is a plane graph, then for each k ∈ N, P (G; k) is equal to the number of admissiblek-valuations of the medial ribbon graph Gm of G.

27

This characterization of the Penrose polynomial does not extend to a general embedded graph, however.

For example, the embedded graph has Penrose polynomial −λ3 + 4λ2 − 3λ, but has k3 − 2k2 − k

admissible k-valuations, for k ≥ 3.

We are now in a position to be able to state and prove our generalization of Aigner’s inequality χ(G∗; k) ≤P (G; k) which relates the chromatic and Penrose polynomials.

Theorem 54. Let G = (V (G), E(G)) be a plane graph, then

P (G;λ) =∑

A⊆E(G)

χ((Gτ(A))∗;λ).

Proof. If H is a ribbon graph, we define a admissible k-boundary valuation ϕ to be a k-colouring ϕ :{1, 2, . . . , k} → BC(H) of the boundary components of H such that if two boundary components share acommon edge then they are assigned different colours. This corresponds to a proper face-colouring of Hwhen H is viewed as cellularly embedded, and hence to a proper coloring of H∗. Thus, the number ofadmissible k-boundary valuations of a ribbon graph H is equal to χ(H∗; k), for k ∈ N.

Each k-admissible valuation of Gm induces a Penrose state of Gm. The Penrose state is constructed byassigning a crossing vertex state of Gm if there is a crossing state at the vertex in the admissible k-valuation,and assigning a white split state otherwise.

Note from Figure 4 that the edges of Gm follow precisely the boundaries of G if we choose a crossingstate at vertices of Gm corresponding to twisted edges of G and a white split state at vertices correspondingto untwisted edges of G. Thus an admissible edge coloring of Gm corresponds exactly to an admissibleboundary coloring of Gτ(A), where A is the set of edges corresponding to vertices of Gm where the localcoloring configuration of the incident edges give a crossing state.

Now let G be a plane graphs. By Theorem 53 we know that for all k ∈ N, P (G; k) is equal to the numberof admissible k-valuations. This is equal to the sum over all Penrose states of admissible k-valuations whichinduce the given Penrose state. Thus,∑

s∈P(Gm)

(number of admissible k-valuations inducing s)

=∑

A⊆E(G)

(number of admissible boundary k-valuations of Gτ(A))

=∑

A⊆E(G)

χ((Gτ(A))∗; k).

Since this holds for all natural numbers k, it follows that the polynomials P (G;λ) and∑A⊆E(G) χ((Gτ(A))∗;λ)

are equal, as required.�

This theorem can be used to reformulate the four colour theorem.

Corollary 55. The following statements are equivalent:

(i) the four colour theorem is true;(ii) for every connected, bridgeless plane graph G there exits A ⊆ E(G) such that χ((Gτ(A))∗; 3) 6= 0;

(iii) for every connected, bridgeless plane graph G there exits A ⊆ E(G) such that χ((Gτ(A))∗; 4) 6= 0;

Proof. Corollary 9 of [2] states that that four colour theorem is equivalent to showing that P (G; 3) > 0 orP (G; 4) > 0 for all connected, bridgeless plane graphs G. Since for all k ∈ N and graphs G′, χ(G′; k) ≥ 0,Theorem 54 tells us that P (G; k) > 0 if and only if one of the summands χ((Gτ(A))∗; k) 6= 0. The resultthen follows. �

28

7. The topochromatic polynomial

In this section we will show that the behaviour of the topological transition polynomial under the twistedduality operation from Section 5 provides a framework for understanding the behaviour of the topologicalTutte polynomial under partial duality. (The topological Tutte polynomial, also known as the ribbon graphpolynomial or Bollobas-Riordan polynomial, is an extension of the Tutte polynomial to embedded graphsintroduced by Bollobas and Riordan in [9, 10]. Its multivariate generalization, the topochromatic polynomialwas introduced in [55].) The behaviour of the topological Tutte polynomial under partial duality is significantbecause of its knot theoretical applications (since partial duality intertwines various recent realizations ofthe Jones polynomial of a (virtual) link (see [41, 47]) as evaluations of the topological Tutte polynomial).We will begin by giving a brief outline of the role that partial duality plays in knot theory.

Seminal results of Thistlethwaite [67] and Kauffman [45] that relate the Tutte, dichromatic and Jonespolynomials were extended in two different ways by Chmutov and Pak in [17] and Dasbach et. al. in [20].In [17] it was shown that the Jones polynomial of checkerboard colourable virtual link is an evaluation ofthe topochromatic polynomial of its signed Tait graph. Using a different construction, in [20] it was shownthat the Jones polynomial is an evaluation of the topological Tutte polynomial of an associated (unsigned)embedded graph. Although both of these results generalize Thistlethwaite’s connection between the Tutteand Jones polynomials, they do so in very different ways. In [56], the first author unified these relationsbetween graph and knot polynomials through an “unsigning” procedure on the Tait graph of a link. This“unsigning” is a special case of the subsequently defined partial duality operation on embedded graphs.A third extension of Thistlethwaite’s result was given by Chmutov and Voltz in [18], where the Jonespolynomial of an arbitrary virtual link was shown to be an evaluation of the topochromatic polynomial.In [19], Chmutov introduced the partial duality operation and studied the behaviour of the (2-variable)topochromatic polynomial under this operation. He then went on to show how this behaviour under partialduality provides a framework which unifies all of these new relations between knot and graph polynomials.

In this section we will explain how twisted duality and the transition polynomial provide a natural, unifiedframework for these recently discovered connections between graph theory and knot theory. We will show thata partial duality relation for the topochromatic polynomial arises naturally from the transition polynomial.We will also show that Vignes-Tourneret’s duality relation for the multivariate signed Bollobas-Riordanpolynomial from [75] follows from our relation.

7.1. The topochromatic polynomial. The topochromatic polynomial is simply a shift (albeit with edgeweights) of the topological Tutte polynomial of Bollobas and Riordan, just as the dichromatic polynomialZ(G; a, b) :=

∑H⊆G a

k(H)be(H) is a shift of the Tutte polynomial:

T (G;x, y) = (x− 1)−k(G)(y − 1)−v(G)Z(G; (x− 1)(y − 1), (y − 1)).

It is for this reason that we use the name “topochromatic polynomial” here. The topochromatic polynomialwas first defined in [55] as a multivariate generalization of the Bollobas and Riordan’s topological Tuttepolynomial from [9, 10]. The generalization is in the spirit of the extensions of the Tutte polynomial to edgeweighted graphs from [77], [8] and [64]. The introduction of the topochromatic polynomial was necessitatedby some of the knot theoretical applications in considered in [55]. The topochromatic polynomial has sincefound other applications, such as in [35], where it was used in to study the behaviour of the topological Tuttepolynomial under the 2-sum operation.

In the literature the topochromatic polynomial is usually applied to the ribbon graph realization of anembedded graph. Here we will generally avoid fixing a realization of an embedded graph. We first establishsome notation. If G is an embedded graph then w(G) = 0 if G is embedded in an orientable surface (orequivalently, the ribbon graph is an orientable surface) and w(G) = 1 if G is embedded in a non-orientablesurface (or equivalently, the ribbon graph is a non-orientable surface). The notion of a spanning ribbonsubgraph is clear, and we define a spanning subgraph of an embedded graph to be the embedded graphcorresponding to a spanning sub ribbon graph. (This ensures that the spanning subgraphs of a cellularlyembedded graph are also cellularly embedded, although not necessarily in the same surface as the originalgraph.)

29

Definition 56. Let G be an embedded graph and b := {be|e ∈ E(G)} be a set of indeterminates indexedby E(G). The topochromatic polynomial is

Z(G; a, b, c, w) =∑H⊆G

ak(H)

∏e∈E(H)

be

cf(H)wt(H) ∈ Z[a, b, c, w]/〈w2 − w〉,

where the sum is over all (embedded) spanning subgraphs H of G.

The topochromatic polynomial contains both the topological Tutte polynomial of Bollobas and Riordan,the classical Tutte polynomial and the restricted normal form of the multivariate Tutte polynomial asspecializations (see [55] for details). We will see below that it is also equivalent to the signed multivariateBollobas-Riordan polynomial from Vignes-Tourneret [75].

It was observed in [35] that the topochromatic satisfies a deletion contraction relation

Z(G; a, b, c, w) = Z(G− e; a, b′, c, w) + beZ(G/e; a, b′, c, w)

where e ∈ E(G) is a non-loop edge and b′ = b − {be}. This deletion-contraction relation allows one thecalculation of the topochromatic in terms of bouquets.

If e and δ(e) are non-loop edges, then we have that Z(G − e; a, b′, c, w) = Z((Gδ(e))/e; a, b′, c, w) and sowe obtain a dual-contraction relation for the topochromatic polynomial:

Z(G; a, b, c, w) = Z((Gδ(e))/e; a, b′, c, w) + beZ(G/e; a, b′, c, w)

The classical Tutte polynomial, T (G;x, y), among many other properties, encodes information aboutfamilies of Eulerian circuits in the medial graph of a plane graph. This theory is the result of a relationbetween the classical Tutte polynomial and the Martin, or circuit partition, polynomial. In [30] this theorywas extended to ribbon graphs, giving an analogous result relating the topological Tutte polynomial ofBollobas and Riordan [9, 10] for a ribbon graph to the transition polynomial of its topological medialgraph, noting that the transition polynomial of [24] is a multivariable generalization of the circuit partitionpolynomial. The original relation between the Tutte polynomial and the Martin polynomial can be foundin Martin’s 1977 thesis [53], with the theory considerably extended by Martin [54], Las Vernas [50, 51, 52],Jaeger [40], Bollobas [11], and [22, 25, 26, 24]. An overview can be found in Ellis-Monaghan and Merino [28,29]. These ideas can be extended to the topochromatic polynomial, as was done for full duality in theunweighted case in [30, 27], and see also [55, 57] and Chmutov [19].Here we extend them to multivariablesgeneralizatins and partial duality. We are particularly interested in an extension of the relation between thetopological Tutte polynomial and the transition polynomial which will allow us to apply our previous resultson twisted duality to the topochromatic polynomial.

Let G be an embedded graph and Gm be its embedded medial graph equipped with the canonical checker-board colouring. We define the weight system WZ(Gm) by

WZ(Gm) :ve

= bv +1 .

The following proposition expresses the topochromatic polynomial and the transition polynomial usingthe weight system WZ .

Proposition 57. If G is an embedded graph and Gm is its embedded medial graph, then

Q (G; (b,1,0), c) = Z(G; 1, b, c, 1).

Proof. By definition,

Q (G; (b,1,0), c) =∑s

ωZ(s)cc(s) =∑s

∏ve∈Wh(s)

be

cc(s),

where the sum is over all graph states s with no crossing states and where Wh(s) is the set vertices withwhite split states in the graph state s.

30

We can define a bijection between the set of embedded spanning subgraphs of G and the set of graphstates of Gm by associating an embedded spanning subgraph Hs of G by setting e ∈ Hs if and only if thevertex state ve ∈ Wh(s). It is then clear that, for every graph state, c(s) = f(Hs). By using this bijection,we have

Q (G; (b,1,0), c) =∑s

∏ve∈Wh(s)

be

cc(s) =∑H⊆G

∏e∈E(H)

be

cf(H) = Z(G; 1, b, c, 1).

�

7.2. Partial duality and the topochromatic polynomial. In this section we use the relation betweenthe topochromatic and transition polynomials of Theorem 57 and the behaviour of the transition polynomialunder twisted duality from Theorem 41 to find a partial duality relation for the topochromatic polynomial.

Let G be an embedded graph and let Gm be its embedded medial graph equipped with the canonicalcheckerboard colouring. Then for A ⊆ E(G), from Definition 40 the weight system W

δ(A)Z (Gm) is given by

reversing the roles of be and 1 if e is in A, thus:

Wδ(A)Z (Gm) :

if e /∈ A thenve

= be +

if e ∈ A thenve

= +be

.

Lemma 58. Let G be an embedded graph with embedded medial graph Gm. Then if A ⊆ E(G), we have

Q(G; (b,1,0)δ(A), c

)=

∏e∈E(G)e∈A

be

Z(G; 1, bA, c, 1),

wherebA := {be | e /∈ A} ∪ {1/be | e ∈ A}.

Proof. The proof is similar to the proof of Proposition 57. We have

Q(G; (b,1,0)δ(A), c

)=∑s

(ωZ)δ(A)(s)cc(s) =∑s

∏ve∈Wh(s)

e/∈A

be

∏ve /∈Wh(s)

e∈A

be

cc(s),

where the sum is over all graph states s with no crossing states and where Wh(s) is the set vertices withwhite split states in the graph state s.

We can define a bijection between the set of embedded spanning subgraphs of G and the set of graph statesof Gm by associating a spanning subgraph Hs of G by setting e ∈ Hs if and only if the vertex ve ∈ Wh(s).It is then clear that for every graph state c(s) = f(Hs). By using this bijection, we can write the above statesum as

=

∏e∈E(G)e∈A

be

∑H⊆G

∏e∈E(H)e/∈A

be

∏e∈E(H)e∈A

1be

cf(H) =

∏e∈E(G)e∈A

be

Z (G; 1, bA, c, 1)

wherebA := {be | e /∈ A} ∪ {1/be | e ∈ A}

as required. �

Our partial duality relation is given in Theorem 59, which is a a multivariable and partial duality extension,in terms of the topochromate as opposed to the topological Tutte polynomial, of the duality relation in ??.

31

Theorem 59. Let G be an embedded graph withA ⊆ E(G). Then

Z(G; 1, b, c, 1) =

(∏e∈A

be

)Z(Gδ(A); 1, bA, c, 1),

whereb = {be|e ∈ E(G)} and bA = {be | e /∈ A} ∪ {1/be | e ∈ A}.

Proof. We have(∏e∈A

be

)Z(Gδ(A); 1, bA, c, 1) = Q

(Gδ(A); (b,1,0)δ(A), c

)= Q (G; (b,1,0), c) = Z(G; 1, b, c, 1),

where the first equality is by Lemma 58, the second is by Theorem 41 and the third follows from Proposi-tion 57. �

The signed Bollobas-Riordan polynomial was introduced by Chmutov and Pak in [17] to extend somerelations between the topological Tutte polynomial and the Jones polynomial of a virtual link, and thenVignes-Tourneret [75] provided a multivariate generalization of the signed Bollobas-Riordan polynomial. Inthe remainder of this section we relate the duality relation in Theorem 59 to Vignes-Tourneret’s dualityrelation.

A signed embedded graph is an embedded graph such that each edge is equipped of a sign + or −. Welet E±(G) denote the set of ± signed edges of G, and e±(G) = |E±(G). Also, if H is an embedded spanningsubgraph of G, H is defined to be the complementary spanning subribbon graph H = (V (G), E(G)−E(H)),and s(H) = 1

2 (e−(H)− e−(H).If G is a signed embedded graph and A ⊆ E(G) , then the partial dual Gδ(A) should also be regarded as

a signed embedded graph. Signs are assigned to the edges of Gδ(A) using the signs of the edges of G in thefollowing way: if e /∈ A, then retain the sign of the edge e; and if e ∈ A, then change the sign of the edge e.

Definition 60. The multivariate signed Bollobas-Riordan polynomial is

Z(G; q,α, c) =∑H⊆G

qk(H)+s(H)

∏e∈E+(H)∪E−(H)

αe

cf(H).

Various properties of the multivariate signed Bollobas-Riordan polynomial were shown by Vignes-Tourneret[75] including its invariance under partial duality of signed embedded graphs. Below we will show that themultivariate signed Bollobas-Riordan polynomial Z is a reformulation of the topochromatic polynomial Z andthat the partial duality invariance of Z follows from Theorem 59. We also note that conversely, Theorem 59follows from the partial duality relation for Z given in [75].

The polynomials Z and Z are actually equivalent up to a prefactor.

Lemma 61. Let G be an embedded graph, then

Z(G; q,α, c) =

∏E−(G)

q−1/2αe

Z(G; q,β, c, 1)

where β = {αe | e ∈ E+(G)} ∪ {qα−1e | e ∈ E−(G)}.

The proof of the lemma is very similar to the rewriting of the signed Ribbon graph polynomial in Section 4.1of [57].

Proof. We have

s(H) =12

(e−(H)− e−(H)) =12

(e−(H)− e(G) + e+(G) + e−(H)) = (e−(H)− 12e−(G)),

so

(10) qs(H) = q(e−(H)− 12 e−(G)).

32

Also E−(H) = E−(G)− E−(H), giving

(11)∏

e∈E−(H)

αe =

∏e∈E−(G)

αe

∏e∈E−(H)

α−1e

.

Therefore

Z(G; q,α, c) =∑H⊆G

qk(H)+s(H)

∏e∈E+(H)

αe

∏e∈E−(H)

αe

cf(H)

=∑H⊆G

qk(H)+s(H)

∏e∈E+(H)

αe

∏e∈E−(G)

q−1/2αe

∏e∈E−(H)

qα−1e

cf(H),

where the second equality follows by equations (10) and (11).Finally, the state sum above is just

(∏e∈E−(G) q

−1/2αe

)Z (G; q,β, c, 1) where βe = αe if e ∈ E+(G) and

βe = qα−1e if e ∈ E−(G). �

We now prove the partial duality result of Vignes-Tourneret [75] as a corollary of Theorem 59.

Corollary 62. Let G be an embedded graph. Then

Z(G; 1,α, c) = Z(Gδ(A); 1,β, c)

Proof of Corollary 62. By Lemma 61 we have

(12) Z (G; 1,α, c) =

∏e∈E−(G)

αe

Z (G; 1, β, c, 1) ,

whereβ = {αe | e ∈ E+(G)} ∪ {α−1

e | e ∈ E−(G)}.Theorem 59 then gives

(13) Z (G; 1, β, c, 1) =

(∏e∈A

βe

)Z(Gδ(A); 1,βA, c, 1

),

where

βA = {αe | e ∈ E+(G), e /∈ A}∪{α−1e | e ∈ E−(G), e /∈ A}∪{α−1

e | e ∈ E+(G), e ∈ A}∪{αe | e ∈ E−(G), e ∈ A}.Using the facts that E±(G)∩A = E∓(Gδ(A))∩A and E±(G)∩A = E±(Gδ(A))∩A, where A is the complementof A in E(G), we can rewrite βA as

βA = {αe | e ∈ E+(Gδ(A))} ∪ {α−1e | e ∈ E−(Gδ(A))}.

Using this rewriting of β, Lemma 61 gives

Z(Gδ(A); 1,α, c) =

∏e∈E−(Gδ(A))

αe

Z(Gδ(A); 1,βA, c, 1

).

Together with Equations (12) and (13), the above identity tells us that

(14) Z (G; 1,α, c) =

∏e∈E−(G)

αe

(∏e∈A

βe

) ∏e∈E−(Gδ(A))

α−1e

Z(Gδ(A); 1,α, c)

To prove the corollary, it remains to show that the factor on the right hand side of (14) is 1. ∏e∈E−(G)

αe

(∏e∈A

βe

)=

∏e∈E−(G)e∈A

αe

∏e∈E−(G)e/∈A

αe

∏e∈E+(G)e∈A

αe

∏e∈E−(G)e∈A

α−1e

33

=

∏e∈E−(G)e/∈A

αe

∏e∈E+(G)e∈A

αe

=

∏e∈E−(Gδ(A))

e/∈A

αe

∏e∈E−(Gδ(A))

e∈A

αe

=∏

e∈E−(Gδ(A))

αe.

The result then follows from Equation (14). �

We note that further analogues of the properties of the polynomial Z which were shown in [75], can bededuced for Z, but do not include the details here.

References

[1] L.M. Aldeman, Molecular computation of solutions to combinatorial problems, Science, 266 no. 5187 (1994), 1021-1024.

[2] M. Aigner, The Penrose polynomial of a plane graph, Math. Ann. 307 (1997), no. 2, 173–189.

[3] M. Aigner, Die Ideen von Penrose zum 4-Farbenproblem. Jahresber. Deutsch. Math. -Verein. 102, 43–68 (2000).[4] M. Aigner,The Penrose polynomial of graphs and matroids. In: Hirschfeld, J. W. P. (ed) Surveys in Combinatorics, 2001.

Cambridge University Press, Cambridge (1997).

[5] M. Aigner, H. Mielke, The Penrose polynomial of binary matroids. Monatsh. Math. 131 (2000), no. 1, 1–13.[6] D. Bar-Natan, Weights of Feynman diagrams and the Vassiliev knot invariants, preprint (February 1991),

http://www.math.toronto.edu/∼drorbn/papers.

[7] D. Bar-Natan, On the Vassiliev knot invariants, Topology, 34 (1995) 423-472.[8] B. Bollobas and O. Riordan, A Tutte polynomial for coloured graphs, Comb. Probab. Comput. 8 (1999) 45–93.

[9] B. Bollobas and O. Riordan, A polynomial for graphs on orientable surfaces, Proc. London Math. Soc. 83 (2001), 513-531.[10] B. Bollobas and O. Riordan, A polynomial of graphs on surfaces, Math. Ann. 323 (2002), 81-96.

[11] B. Bollobas, Evaluations of the circuit partition polynomial. J. Combin. Theory Ser. B, 85, 261–268 (2002).

[12] T. Brylawski, Thesis, Dartmouth College, Hanover, New Hampshire, 1970.[13] T. Brylaski, The Tutte polynomial, Part 1: General theory, in A. Barlotti (ed.), Matroid Theory and Its Applications,

Proceedings of the Third Internationsl Mathematical Summer Center (C.I.M.E. 1980), 125-275, 1982.

[14] S. V. Chmutov, S. V. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, draft available fromhttp://www.math.ohio-state.edu/∼chmutov/preprints/.

[15] S. V. Chmutov and S. V. Duzhin, An upper bound for the number of Vassiliev knot invariants, J. Knot Theory Ramifications

3 (1994), no. 2, 141-151.[16] J. Chen and N.C. Seemann, Synthesis from DNA of a molecule with the connectivity of a cube, Nature, 350 (1991), 631-633.

[17] S. Chmutov and I. Pak, The Kauffman bracket of virtual links and the Bollobas-Riordan polynomial, Mosc. Math. J. 7

(2007) 409-418, arXiv:math.GT/0609012.[18] S. Chmutov, J. Voltz, Thistlethwaite’s theorem for virtual links, J. Knot Theory Ramifications 17 (2008), 1189-1198

arXiv:0704.1310 .[19] S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobas-Riordan polynomial, J. Combin. Theory

Ser. B, 99 (2009), 617-638, arXiv:0711.3490.

[20] O. T. Dasbach, D. Futer, E. Kalfagianni, X.-S. Lin, N. W. Stoltzfus, The Jones polynomial and graphs on surfaces, J.Combin. Theory Ser. B, 98 (2) (2008), 384-399 arXiv:math.GT/0605571.

[21] J. Edmonds, On the surface duality of linear graphs, J. Res. Nat. Bur. Standards Sect. B 69B (1965) 121-123.

[22] J. A. Ellis-Monaghan, Differentiating the Martin polynomial, Cong. Num. 142 (2000), 173-183.[23] J. A. Ellis-Monaghan, I. Sarmiento, Medial graphs and the Penrose polynomial. Proceedings of the Thirty-second South-

eastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr.

Numer. 150 (2001), 211–222.[24] J. A. Ellis-Monaghan, I. Sarmiento, Generalized transition polynomials, Congr. Numer. 155 (2002) 57-69.

[25] J. A. Ellis-Monaghan, Identities for the circuit partition polynomials, with applications to the diagonal Tutte polynomial.Advances in Applied Mathematics, 32, 188–197 (2004)

[26] J. A. Ellis-Monaghan, Exploring the Tutte-Martin connection. Discrete Mathematics, 281, 173–187 (2004)

[27] J. A. Ellis-Monaghan and I. Sarmiento, A duality relation for the topological Tutte polynomial, talk at the

AMS Eastern Section Meeting Special Session on Graph and Matroid Invariants, Bard College, 10/9/2005.http://academics.smcvt.edu/jellis-monaghan/#Research

[28] J. A. Ellis-Monaghan, C. Merino, Graph polynomials and their applications I: the Tutte polynomial, in Structural Analysisof Complex Networks, Matthias Dehmer, ed., in press arXiv:0803.3079.

[29] J. A. Ellis-Monaghan, C. Merino, Graph polynomials and their applications II: interrelations and interpretations, in

Structural Analysis of Complex Networks, Matthias Dehmer, ed., in press arXiv:0806.4699 .[30] J. A. Ellis-Monaghan and I. Sarmiento, A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan,

preprint arXiv:0903.2643.

[31] H. Fleischner, Eulerian graphs and related topics, Part 1, Volume I, Ann. Discrete Math. 45 (1990).[32] H. Fleischner, Eulerian graphs and related topics, Part 1, Volume 2, Ann. Discrete Math. 50 (1990).

[33] J. L. Gross, T. W. Tucker, Topological graph theory, Wiley-interscience publication, 1987.

[34] P. Freyd, J. Hoste, W. B. R. Lickorish, K. Millett, A. Ocneanu, and D. Yetter, A new polynomial invariant of knots andlinks, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239-246.

34

[35] S. Huggett and I. Moffatt , Expansions for the Bollobs-Riordan and Tutte polynomials of separable ribbon graphs, Annalsof Combinatorics, in press arXiv:0710.4266.

[36] F. Jaeger, Tutte polynomials and link polynomials, Proc. Amer. Math. Soc. 103 (1988), no. 2, 647-654.

[37] F. Jaeger, On Tutte polynomials and cycles of plane graphs. J. Combin. Theory Ser. B 44 (1988), no. 2, 127–146.[38] F. Jaeger, On the Penrose number of cubic diagrams. Graph colouring and variations. Discrete Math. 74 (1989), no. 1-2,

85–97.

[39] F. Jaeger, On transition polynomials of 4-regular graphs. Cycles and rays (Montreal, PQ, 1987), 123–150, NATO Adv. Sci.Inst. Ser. C Math. Phys. Sci., 301, Kluwer Acad. Publ., Dordrecht, 1990.

[40] F. Jaeger, On transition polynomials of 4-regular graphs, In: Cycles and Rays (Hahn et al, eds.) Kluwer, (1990), 123-150.[41] V. F. R. Jones, Vaughan, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985),

no. 1, 103-111.

[42] N. Jonoska, M. Saito, Boundary components of thickened graphs, revised papers of the 7th International Meeting on DNAbased computers, Eds. N. Jonoska and N.C. Seeman, Springer LNCS, 2340 (2002), 70-81.

[43] N. Jonoska, G. McColm, A. Staninska Spectrum of a pot for DNA complexes, in DNA Computing 12 (editors: C. Mao,

T. Yokomori), Springer LNCS, 4287 (2006), 83-94.[44] N. Jonoska, G. McColm, A. Staninska, Expectation and Variance of Self-Assembled Graph Structures, in DNA Computing

(DNA11), (editors: A. Carbone N.A. Pierce), Springer LNCS, 3892 (2006), 144–157.

[45] L. H. Kauffman, A Tutte polynomial for signed graphs, Combinatorics and complexity (Chicago, IL, 1987). Discrete Appl.Math. 25 (1989), no. 1-2, 105-127.

[46] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.

[47] L. H. Kauffman, Virtual knot theory, Europ. J. Combinatorics 20 (1999) 663-690.[48] T.H. LaBean, H. Li, Constructing novel materials with DNA, nanotoday, 2 no. 2 (2007), 26-35.

[49] M. Loebl and I. Moffatt, The chromatic polynomial of fatgraphs and its categorification, Advances in Mathematics, 217(2008) 1558-1587.arXiv:math/0511557.

[50] M. Las Vergnas, On Eulerian partitions of graphs, Graph Theory and Combinatorics, R. J. Wilson, ed., Research Notes in

Mathematics 34, Pitman Advanced Publishing Program, San Francisco, London, Melbourne, (1979), 62-65.[51] M. Las Vergnas, Le polynome de Martin d’un Graphe Eulerien, Ann. Discrete Math. 17 (1983) 397-411.

[52] M. Las Vergnas, On the evaluation at (3, 3) of the Tutte polynomial of a graph, J. Combin. Thry. Series B, 44 (1988)

367-372.[53] P. Martin, Enumerations Euleriennes dans le multigraphs et invariants de Tutte-Grothendieck, Thesis, Grenoble, 1977.

[54] P. Martin, Remarkable valuation of the dichromatic polynomial of planar multigraphs, Journal of Combinatorial Theory,

Series B, 24 (1978) 318-324.[55] I. Moffatt, Knot invariants and the Bollobas-Riordan polynomial of embedded graphs, European Journal of Combinatorics,

29 (2008) 95-107 arXiv:math/0605466.

[56] I. Moffatt, Unsigned state models for the Jones polynomial, Ann. Comb., in press, arXiv:0710.4152.[57] I. Moffatt, Partial duality and Bollobas and Riordan’s ribbon graph polynomial, preprint arXiv:0809.3014.

[58] I. Moffatt, A characterization of partially dual graphs, preprint arXiv:0901.1868.

[59] J. H. Przytycki and P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988), no. 2, 115-139.[60] R. Penrose, Applications of negative dimensional tensors. 1971 Combinatorial Mathematics and its Applications (Proc.

Conf., Oxford, 1969) pp. 221–244 Academic Press, London.[61] I. Sarmiento, Hopf algebras and the Penrose polynomial. European J. Combin. 22 (2001), no. 8, 1149–1158

[62] A. Staninska, The Graph of a Pot with DNA molecules, Proceedings of the 3rd annual conference on Foundations of

Nanoscience (FNANO’06), April 2006, 222-226.[63] W.M. Shih, J.D. Quispe, G.F. Joyce, A 1.7 kilobase single-stranded DNA that folds into a nanoscale octahedron, Nature,

427 (2004), 618-621.

[64] A. D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. Surveys in combinatorics 2005,173-226, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005. arXiv:math.CO/0503607.

[65] B. Steele, Buckyballs demonstrate DNA as building material, Cornell Chronical, September 1 (2005), 9.

[66] C. Szegedy, On the number of 3-edge colorings of cubic graphs. European J. Combin. 23 (2002), no. 1, 113–120.[67] M. B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297-309.

[68] L. Traldi, A dichromatic polynomial for weighted graphs and link polynomials, Proc. Amer. Math. Soc. 106 (1989), no. 1,279-286.

[69] V. Turaev, A simpe proof of the Murasugi and Kauffman theorems on alternating links, L’Enseignement Mathematique

33 (1987) 203-225.[70] Tutte, W. T.: A ring in graph theory. Proc. Cambridge Phil. Soc., 43, 2640 (1947).

[71] Tutte, W. T.: A contribution to the theory of chromatic polynomials. Can. J. Math., 6, 8091 (1954).

[72] Tutte, W. T.: On dichromatic polynomials. J. Combin. Theory, 2, 301 320 (1967).[73] Tutte, W. T.: Graph Theory. Cambridge University Press, Cambridge (1984).

[74] Tutte,W. T.: All the kings horses. In: Bondy, J. A., Murty U. S. R. (eds) Graph Theory and Related Topics. Academic

Press, London (1979)[75] F. Vignes-Tourneret, The multivariate signed Bollobas-Riordan polynomial, preprint arXiv:0811.1584.

[76] H. Yan, S.H. Park, G. Finkelstein, J. Reif, T. LaBean, DNA-templated self-assembly of protein arrays and highly conductive

nanowires, Science, 301 (2003), 1882-1884.

35

[77] T. Zaslavsky, Strong Tutte functions of matroids and graphs, Trans. Amer. Math. Soc. 334 (1992) 317–347.[78] Y. Zhang, N.C. Seeman, Construction of a DNA-truncated octahedron, J. Am. Chem. Soc., 116 (1994), 1661-1669.

36