On Alexander Polynomials of Graphsmath.sjtu.edu.cn/conference/Bannai/2018/data/20180615B/...Alexander polynomials The Alexander polynomial of links was rst studied by J. W. Alexander

On Alexander Polynomials of Graphs

Zhongtao Wu(joint with Yuanyuan Bao)

The Chinese University of Hong Kong

June 15, 2018

Zhongtao Wu (joint with Yuanyuan Bao) (CUHK)On Alexander Polynomials of Graphs June 15, 2018 1 / 20

Alexander polynomials

The Alexander polynomial of links was first studied by J. W. Alexander in1920s, which is a very useful and powerful invariant.

Later people found many different ways of definitions:

universal abelian cover of link complements

Seifert surfaces

Kauffman’s state sum formula

Conway’s skein relations

and many more ...






Seifert surfaces



and many more ...






Seifert surfaces



and many more ...


Spatial graph theory

We study embedded graphs in this talk.

A graph is a finite collection of vertices V together with disjoint edgesE connecting pairs of vertices.

An embedded graph G means a graph that exists in a specificposition in the three-space, whereas an abstract graph g is a graphthat is considered to be independent of any particular embedding.

Spatial graph theory is the study of embedded graphs in the space.




















MOY graphs

An MOY graph is an embedded graph equipped with

a transverse orientation

...

...

1

2

...

...v

v

v

Lv

a balanced coloring c : E → Z≥0 such that for each vertex v ∈ V ,

∑

incoming

c(e) =∑

outgoing

c(e).


MOY graphs

A closed MOY graph is an MOY graph without vertex of valence one.

Example

(Singular) knots/links

Embedded Θ graphs


MOY graphs

A closed MOY graph is an MOY graph without vertex of valence one.

Example

(Singular) knots/links

Embedded Θ graphs


Alexander polynomials of spatial graphs

One can define an Alexander polynomial ∆(G ,c)(t) ∈ Z[t]/± tn for closedMOY graphs via the following standard method in covering space.

coloring c determines a homomorphism

φc : π1(S3 − G )→ H1(S3 − G ;Z)→ Z〈t〉.

Denote X = S3 − G , and let ∂inX be a subsurface of ∂X . Let X bethe cyclic covering of X corresponding to ker(φc).

The deck transformation endows the relative homologyH1(X , p−1(∂inX )) with a Z[t−1, t]-module structure. Call this theAlexander module associated to (G , c).

Define ∆(G ,c)(t) to be the determinant of a presentation matrix ofthe Alexander module.


Alexander polynomials of spatial graphs

One can define an Alexander polynomial ∆(G ,c)(t) ∈ Z[t]/± tn for closedMOY graphs via the following standard method in covering space.

coloring c determines a homomorphism

φc : π1(S3 − G )→ H1(S3 − G ;Z)→ Z〈t〉.

Denote X = S3 − G , and let ∂inX be a subsurface of ∂X . Let X bethe cyclic covering of X corresponding to ker(φc).

The deck transformation endows the relative homologyH1(X , p−1(∂inX )) with a Z[t−1, t]-module structure. Call this theAlexander module associated to (G , c).

Define ∆(G ,c)(t) to be the determinant of a presentation matrix ofthe Alexander module.


A second definition via Kauffman state sum

Alternatively, ∆(G ,c)(t) can be defined in a more combinatorial andconcrete manner.

Starting with a graph projection/diagram D of G , draw a circle aroundeach vertex.

Cr(D): set of crossings

Re(D): set of regions

Lemma

|Re(D)| = |Cr(D)|+ 2 if D is a connected graph diagram.







Lemma








Lemma



Procedure of computing Kauffman state sum

Mark a point δ on an edge and the two nearby regions.

A state is a bijective map s : Cr(D)→ Re(D)\{marked regions}.Define local contributions M4c and A4c .

... ...

-1

11

1

1

1

1

-1

1 1

1

1

1 1

1

... ...

t i t

t -i/2tt

t - ti/2

i/2 -i/2

t i

i i

i

-i

-i


Procedure of computing Kauffman state sum

Sum over all states

∑

s

∏

crossing c

Ms(c)c A

s(c)c

Multiply the above sum with (a factor depending on δ) and (anotherfactor depending on D) to get a graph invariant.

Theorem

The above invariant coincides with the Alexander polynomial ∆(G ,c)(t)defined earlier using cyclic covering; furthermore, it resolves the ±tnambiguity and gives a normalized Alexander polynomial.


An example of θ-curve

The 51 in Litherland’s table of θ-curve diagrams

i+jj

i

*δ

e

d

cb

a4

e

1dd

d

c

bb a

1

1

1

12

2

2

2

23

3

There are Kauffman states: a1b1c1d2e1, a1b1c1d3e2, a1b1c2d4e2,a2b2c1d2e1, a2b2c1d3e2, a2b2c2d4e2, a2b3c2d1e2


An example of θ-curve

The 51 in Litherland’s table of θ-curve diagrams

i+jj

i

*δ

e

d

cb

a4

e

1dd

d

c

bb a

1

1

1

12

2

2

2

23

3

There are Kauffman states: a1b1c1d2e1, a1b1c1d3e2, a1b1c2d4e2,a2b2c1d2e1, a2b2c1d3e2, a2b2c2d4e2, a2b3c2d1e2


Calculations of the Alexander polynomial

Compute each term of the state sum

∑

s

∏

crossing c

Ms(c)c A

s(c)c .

For example, the contribution of the state a1b1c1d2e1 is

t3i2+2j · (t i+j

2 − t−i+j2 ) · (t j

2 − t−j2 )

After summing up and multiplying with the appropriate factors, weobtain the normalized Alexander polynomial

∆(G ,c)(t) = (t3i+3j

2 − t3i+j2 − t

i+3j2 + t

i+j2 + t

i−j2 + t

j−i2 − t

−i−j2 ) · [i + j ]

where [i + j ] := t(i+j)/2−t−(i+j)/2

t1/2−t−1/2


Calculations of the Alexander polynomial

Compute each term of the state sum

∑

s

∏

crossing c

Ms(c)c A

s(c)c .

For example, the contribution of the state a1b1c1d2e1 is

t3i2+2j · (t i+j

2 − t−i+j2 ) · (t j

2 − t−j2 )

After summing up and multiplying with the appropriate factors, weobtain the normalized Alexander polynomial

∆(G ,c)(t) = (t3i+3j

2 − t3i+j2 − t

i+3j2 + t

i+j2 + t

i−j2 + t

j−i2 − t

−i−j2 ) · [i + j ]

where [i + j ] := t(i+j)/2−t−(i+j)/2

t1/2−t−1/2


MOY graphic calculus

Murakami, Ohtsuki and Yamada developed a graphic calculus forUq(slN)-polynomial invariants in the late 1990s.

Defined for N ≥ 2

N = 2 case: Jones polynomial

N = 0 case: Alexander polynomials


MOY relations for Uq(slN)

THE MODULI PROBLEM OF LOBB AND ZENTNER FOR COLORED SL(N) 5

(Move 0)

(i

)

N

=

[Ni

]

(Move 1)

⎛⎜⎜⎝

i

i

j + i j

⎞⎟⎟⎠

N

=

[N − i

j

]⎛⎜⎜⎝ i

⎞⎟⎟⎠

N

(Move 2)

⎛⎜⎜⎝

i

i

i − j j

⎞⎟⎟⎠

N

=

[ij

]⎛⎜⎜⎝ i

⎞⎟⎟⎠

N

(Move 3)

⎛⎜⎜⎜⎝

i + j + k

k

i + j

i j⎞⎟⎟⎟⎠

N

=

⎛⎜⎜⎜⎝

i + j + k

k

j + k

i j⎞⎟⎟⎟⎠

N

(Move 4)

⎛⎜⎜⎜⎝

1 i

i

1 i

1

i + 1

i + 1⎞⎟⎟⎟⎠

N

= [N − i − 1]

⎛⎜⎜⎜⎝

1 i

i − 1

1 i⎞⎟⎟⎟⎠

N

+

⎛⎜⎜⎜⎝ 1 i

⎞⎟⎟⎟⎠

N

(Move 5)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 i + j − 1

i + k

i

i + k − 1

j − k

k

j⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

N

=

[j − 1k − 1

]

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 i + j − 1

i j

i + j

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

N

+

[j − 1

k

]

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

i

1

j

i + j − 1

i − 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

N

Figure 6. The six MOY moves

In all cases orientations can be chosen arbitrarily, but must be chosen consistently for two sides ofa given move.

Theorem 5.1. The six MOY moves uniquely determine the MOY sl(N) polynomial for colouredoriented trivalent planar graphs.

To prove this, we first specialise to {1, 2}-coloured graphs:

Proposition 5.2. The six MOY moves, specialised to colourings in {1, 2}, determine the sl(N) poly-nomial for oriented trivalent plane graphs coloured with {1, 2}.Proof. Given a diagram coloured in {1, 2}, we can use Move 0 to remove closed loops coloured with2, so suppose the remaining graph Γ has n edges coloured 2. We use the relationship with knotdiagrams in Section 2 to construct a knot diagram D with n crossings for which the resolution withthe most 2-coloured edges is Γ. With appropriate choice of crossings, we can ensure that D is adiagram for an unlink U . By the work of MOY, if the polynomial satisfies the MOY moves it alsosatisfies the Reidemeister moves, so the polynomial is a link invariant. Then (U)N = (−1)n(Γ)N +∑2n−1

i=1 (−1)ki(q)ϵi(Γi)N where each Γi has ki < n edges coloured with 2, and ϵi is the number ofpositive crossings resolved into edges coloured 1 minus the number of negative crossings resolved into


Skein relations for Uq(slN)

2 JONATHAN GRANT

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

N

= q

⎛⎜⎜⎜⎝

1 1

⎞⎟⎟⎟⎠

N

−

⎛⎜⎜⎜⎜⎝

1

1

1

1

2

⎞⎟⎟⎟⎟⎠

N

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠

N

= q−1

⎛⎜⎜⎜⎝

1 1

⎞⎟⎟⎟⎠

N

−

⎛⎜⎜⎜⎜⎝

1

1

1

1

2

⎞⎟⎟⎟⎟⎠

N

Figure 1. MOY resolutions of knot diagrams

⎛⎜⎜⎜⎝

i j⎞⎟⎟⎟⎠

N

=

i∑

k=0

(−1)k+(j+1)iqi−k

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

j i

j + k

i

k

i − k

j + k − i

j⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

N

Figure 2. MOY resolutions of a coloured knot diagram if i ≤ j

⎛⎜⎜⎜⎝

i j⎞⎟⎟⎟⎠

N

=

i∑

k=0

(−1)k+(i+1)jqj−k

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

j i

j − k

i

k

i + k

i + k − j

j⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

N

Figure 3. MOY resolutions of a coloured knot diagram if i > j

3. A Moduli Space of Colourings

In their paper [3] Lobb and Zentner introduced a moduli space M (Γ) of colourings of a diagram Γby associating to an i-coloured edge an element of the complex Grassmannian G(i, N) in such a waythat if the three edges around a vertex are coloured i, j and i+ j, then the i-plane and the j-plane areorthogonal and span the (i + j)-plane in CN . They showed that if Γ is coloured with 1’s and 2’s, then

χ(M (Γ)) = (Γ)N (1)

ie. the Euler characteristic is the MOY polynomial evaluated at 1. It is tempting to think that in factthe Poincare polynomial of M (Γ) is equal to (Γ)N , but Lobb and Zentner showed that this is false ingeneral.

In this paper, we show that the same relation holds for all higher colourings as well.

Theorem 3.1. For a coloured planar trivalent graph Γ, we have

χ(M (Γ)) = (Γ)N (1).


MOY relations for ∆(D,c)(t)

Using the above Kauffman state sum formulation, one can check that∆(D,c)(t) satisfies a set of 10 relations.

Conversely, these 10 relations uniquely determine ∆(D,c)(t) for allclosed MOY graphs. Thus, it may be viewed as a third definition ofAlexander polynomials.

In particular, this gives one more interpretation of the classicalAlexander polynomials of links.












Some properties of ∆(G ,c)(t)

Symmetries: If G ∗ is the mirror image of G , then

∆(G ,c)(t) = ∆(G∗,c)(t−1).

If −G is the graph with opposite orientation, then

∆(G ,c)(t) = ∆(−G ,c)(t).

Positivity: Suppose G is a planar graph of at least 1 vertex equippedwith a non-negative coloring c . Then the non-zero coefficients of theAlexander polynomial ∆(G ,c)(t) are all positive, that is,

∆(G ,c)(t) ∈ Z≥0[t±14 ].



Symmetries: If G ∗ is the mirror image of G , then

∆(G ,c)(t) = ∆(G∗,c)(t−1).

If −G is the graph with opposite orientation, then

∆(G ,c)(t) = ∆(−G ,c)(t).

Positivity: Suppose G is a planar graph of at least 1 vertex equippedwith a non-negative coloring c . Then the non-zero coefficients of theAlexander polynomial ∆(G ,c)(t) are all positive, that is,

∆(G ,c)(t) ∈ Z≥0[t±14 ].



Non-vanishing: If G is a connected planar graph equipped with apositive coloring c , then the Alexander polynomial is non-vanishing:

∆(G ,c)(t) 6= 0.

Intrinsic invariant: Suppose G and G ′ are two different spatialembedding of an abstract directed graph g , then the value of theAlexander polynomial evaluated at t = 1 is the same:

∆(G ,c)(1) = ∆(G ′,c)(1).

In other words, ∆(g ,c) := ∆(G ,c)(1) is an intrinsic invariant of anabstract graph g .



Non-vanishing: If G is a connected planar graph equipped with apositive coloring c , then the Alexander polynomial is non-vanishing:

∆(G ,c)(t) 6= 0.

Intrinsic invariant: Suppose G and G ′ are two different spatialembedding of an abstract directed graph g , then the value of theAlexander polynomial evaluated at t = 1 is the same:

∆(G ,c)(1) = ∆(G ′,c)(1).

In other words, ∆(g ,c) := ∆(G ,c)(1) is an intrinsic invariant of anabstract graph g .


Relation with Viro’s gl(1|1)-Alexander polynomial

In 2006, Viro defined an Alexander polynomial ∆(G , c) for framedgraphs via refinements of Reshetikhin-Turaev functors based onirreducible representations of quantized gl(1|1).

Bao observed that ∆(G , c) satisfies a set of relations that are nearlyidentical to the MOY relations for our Alexander polynomial ∆(G , c).Consequently, the two Alexander polynomial invariant are essentiallythe same.


Relation with Viro’s gl(1|1)-Alexander polynomial

In 2006, Viro defined an Alexander polynomial ∆(G , c) for framedgraphs via refinements of Reshetikhin-Turaev functors based onirreducible representations of quantized gl(1|1).

Bao observed that ∆(G , c) satisfies a set of relations that are nearlyidentical to the MOY relations for our Alexander polynomial ∆(G , c).Consequently, the two Alexander polynomial invariant are essentiallythe same.


Future work

In a work in progress, we define a Heegaard Floer homology for MOYgraphs. We expect to construct a homology theory that satisfies thefollowing properties.

The homology HFG−d (G , s) is bigraded with a Maslov grading d andan Alexander grading s, both depending on the color c .

For a planar graph G , the group HFG−(G ) is determined by theAlexander polynomial ∆(G ,c)(T ); indeed, the homology HFG−d (G , s)is supported on the line 2s = d .

In general, the Euler characteristic gives the Alexander polynomial, inthe sense that:

∑

d ,s

(−1)ddim(HFG−d (G , s)) · ts = ∆(G ,c)(t).


Future work





∑

d ,s



Future work





∑

d ,s



Future work





∑

d ,s



The End

Thank you.


Documents

On Alexander Polynomials of Graphsmath.sjtu.edu.cn/conference/Bannai/2018/data/20180615B/...Alexander polynomials The Alexander polynomial of links was rst studied by J. W. Alexander