The Matiyasevi h polynomial, four olour

theorem and weight systems

Sergei Duzhin

Started on 00.01.06. This version 00.01.06

Abstra t

We show that a modi� ation of the polynomial introdu ed by

Yu. Matiyasevi h for sphere triangulations, an be de�ned to be an

invariant of 3-graphs and thus provide a weight system for �nite type

knot invariants. The relation of this onstru tion with the four- olour

theorem is also dis ussed.

1 Weight systems

Weight systems are ru ial in the ombinatorial study of �nite type (Vas-

siliev) knot invariants. They are de�ned as fun tions on the spa es generated

by graphs with all verti es of valen y 1 or 3, satisfying ertain relations. For

the sake of this exposition, I will give a pre ise de�nition of a weight system

on the spa e of 3-graphs (see [CDK℄). Relation of the spa e of 3-graphs to

other spa es of diagrams is lari�ed in [CDK℄.

De�nition 1 A 3-graph is a regular 3-valent graph

1

with a �xed rotation.

The rotation is the hoi e of a y li order of edges at every vertex, i. e.

one of the two y li permutations in the set of three edges adja ent to this

vertex.

1

Note that the number of verti es of su h a graph is always even, and the number of

edges a multiple of 3. Half the number of the verti es is referred to as the degree of a

3-graph.

1

De�nition 2 The spa e �

n

is the quotient spa e of the linear spa e over Q

generated by onne ted 3-graphs of degree n (i. e. having 2n verti es), modulo

the following relations.

AS (antisymmetry) relation:

�

�

�Z

Z

Z

q

i

�

�

�Z

Z

Z

q

�

=

IHX relation:

�

�

�P

P

P

P

P

P�

�

�

q

q

�

�

�

�

�

�B

B

B

B

B

B

q q

�

�

�

�

��

�

�

�

�

q q

= �

De�nition 3 A weight system of degree n is a fun tion on the spa e �

n

with

values in some Abelian group. In other words, a weight system is a fun tion

of a 3-graph whi h satis�es the AS and IHX relations.

A well known onstru tion of weight systems is Kontsevi h's onstru tion

of the invariant with values in the universal enveloping algebra of a Lie al-

gebra equipped with a �xed ad-invariant non-degenerate symmetri bilinear

form (see [Kon℄ for the original onstru tion, [BN1℄ for the spe ialization to

linear representations and [CDK℄ for the ase of the algebra �). In this on-

stru tion, one asso iates a opy of the stru ture tensor of the Lie algebra g

to every vertex and then makes the ontra tion over all edges. The stru ture

tensor, moved into the spa e g g g by means of the metri , is a totally

antisymmetri element of this spa e, whi h ensures the ompatibility with

the AS relation. The IHX relation follows from the Ja obi identity.

2 Matiyasevi h's polynomial

Let � : V (G) ! f1; 2; : : : ; 3ng be a numbering of the set of edges of the

graph G. We assign an independent variable x

i

, i = 1; : : : ; 3n, to the edge

number i and, with every vertex v 2 V (G), we asso iate the polynomial

(v

1

� v

2

)(v

2

� v

3

)(v

3

� v

1

); (1)

2

if v

1

, v

2

, v

3

are the variables assigned to the three edges meeting at v, taken

in the order onsistent with the rotation at v. Set

M

�

(G) =

Y

v2V (G)

(v

1

� v

2

)(v

2

� v

3

)(v

3

� v

1

):

This is the numbered Matiyasevi h's polynomial. To obtain an invariant

obje t, symmetrize M

�

(G) over all numberings �, or over all permutations

of x

1

, ..., x

3n

:

M(G) =

1

(3n)!

X

�2Perm(3n)

Y

v2V (G)

(�(v

1

)� �(v

2

))(�(v

2

)� �(v

3

))(�(v

3

)� �(v

1

)):

Theorem 1 The Matiyasevi h's polynomial M : �

n

! SQ [x

1

; : : : ; x

3n

℄ is a

weight system on the spa e �

n

with values in the spa e of symmetri polyno-

mials in 3n variables.

Proof. The AS relation, as well as the orre tness of the de�nition of M,

follow from the fa t that expression (1) is totally antisymmetri with respe t

to the permutations of v

1

, v

2

and v

3

. The IHX relation is a onsequen e of

the following remarkable polynomial identity:

(a� b)( � d) + (b� )(a� d) + ( � a)(b� d) = 0: (2)

Relation with the so(3) weight system. In general, a weight system for

3-graphs an be onstru ted from any obje t whi h is skew- ommutative

in 3 variables. Both in the ase of so(3) and in the ase of Matiyasevi h,

we assign a ertain element of R

3

R

3

R

3

to every vertex of the graph,

and this element is totally antisymmetri with respe t to some a tion of the

permutation group on 3 symbols. But the group a tions in question are

di�erent:

� In the so(3) ase, the spa e R

3

is identi�ed with so(3) and the group

Perm(3) a ts in

R

3

R

3

R

3

by permutations of the three fa tors of the tensor produ t.

� In Matiyasevi h's ase, R

3

is the linear span of the formal variables v

1

,

v

2

, v

3

, while the group Perm(3) a ts in

R

3

R

3

R

3

by the same permutaion of the bases in ea h of the three fa tors.

3

Remark. Polynomial identity 2 is the analogue of Ja obi identity in

some sense whi h still needs to be understood.

3 Four olour theorem

De�nition 4 An edge 3- olouring of a three-valent graph is a olouring of

the set of its edges with three olours, su h that the three edges meeting at

any vertex have di�erent olours.

De�nition 5 The number of olourings of a 3-graph G is de�ned as

�(G) =

X

edge

3- olourings of G

Y

v2V

�

v

;

where �

v

is 1 or �1, depending on whether the order of the three olours

meeting at v agrees with the rotation at v.

The famous four olour theorem is equivalent to the assertion that �(G) 6=

0 for any planar 3-graph G. As explained in [BN2℄ and [CDK℄, the number

�(G) is equal to the value of the so(3) weight system on G.

To explain the relation of the Matiyasevi h's polynomial to the four olour

theorem, we introdu e the redu ed Matiyasevi h's polynomial RM(G) as fol-

lows:

RM(G) = M(G)j

x

3

i

7!1

;

In other words, RM(G) is obtained from M(G) by redu ing all the exponents

modulo 3.

Theorem 2 The onstant term of the redu ed Matiyasevi h polynomial is

equal to the number of edge olourings of the 3-graph:

RM(G)j

x

i

=0

= �(G):

Proof. We will prove that the onstant term of every numbered re-

du ed Matiyasevi h's polynomial (before symmetrization) is equal to �(G).

Indeed, for any edge numbering �, the polynomial M

�

(G) is the produ t of

polynomials

v

2

1

v

2

+ v

2

2

v

3

+ v

2

3

v

1

� v

2

2

v

1

� v

2

3

v

2

� v

2

1

v

3

:

4

The hoi e of a term in every su h polynomial is equivalent to a olouring

of the set of half-edges of the given graph by three olours: 0, 1 and 2. For

example, the term v

2

1

v

2

means putting a 2 on the half-edge orresponding

to v

1

, a 1 on the half-edge orresponding to v

2

and a 0 on the half-edge

orresponding to v

3

. Contributions to the onstant term of M

�

(G) arise from

su h olourings where the two ends of every edge either have olours f0; 0g

or olours f1; 2g. We laim that the algebrai number of su h olourings is

equal to the algebrai number of onventional edge 3- olourings, where both

endpoints of every edge have the same olour. Indeed, take an half-edge 3-

olouring and delete all edges of olour 0. What remains is the disjoint union

of several y les. If all the y les are of even length, then, for ea h of them,

there are exa tly two edge olourings and exa tly two half-edge olourings,

and all the signs agree. If there is at least one y le of odd length, then

there is no edge olouring, while the half-edge olourings appear in mutually

an elling pairs.

4 Open problems

1. Is the polynomial M independent of lassi al Lie algebra weight sys-

tems?

2. Is it possible to dete t knot inversion by polynomial M? In other words,

is there a Chinese hara ter (in the sense of [BN1℄) with an odd number

of univalent verti es whose M-polynomial is di�erent from zero?

3. The polynomial M is related to Lie algebra so(3) in the sense des ribed

above. Are there any generalizations of M for other Lie algebras?

5 A knowledgments

I thank S. Chmutov for permanent fruitful dis ussions and S. Me hveliani for

his help in omputations using omputer algebra system DoCon2 ([DoC℄). I

am also indebted to T. Kohno for en ouraging me to prepare this paper for

the \Art of Low Dimensional Topology VI", Kyoto, January 2000.

5

Referen es

[BN1℄ D. Bar-Natan, On the Vassiliev knot invariants. Topology, 34 (1995)

423{472. http://www.ma.huji.a .il/~drorbn/LOP.html

[BN2℄ D. Bar-Natan, Lie algebras and the four olor theorem. Combinatori a

17-1 (1997) 43-52. http://www.ma.huji.a .il/~drorbn/LOP.html

[CDK℄ S. Chmutov, S. Duzhin, A. Kaishev, The algebra of 3-graphs.

Trans. Steklov Math. Institute, vol. 221 (1998), pp.168{196.

http://ftp.botik.ru/pub/lo al/zmr/duzh-pap.html

[Kon℄ M. Kontsevi h, Vassiliev's knot invariants, Adv. in Soviet Math., 16

Part 2 (1993) 137{150.

[YuM℄ Yu. Matiyasevi h, A Polynomial related to Colourings of Tri-

angulation of Sphere. Personal Journal of Yu. Matiyasevi h.

http://logi .pdmi.ras.ru/~yumat/Journal/journal.htm

[DoC℄ S. Me hveliani, The Algebrai Domain Constru tor DoCon,

version 2. Software and manual. Pereslavl-Zalessky, 1999.

http://http.botik.ru/pub/~me hvel

e-mail duzhin�botik.ru

WWW http://www.botik.ru/~duzhin

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