On the universal Vassiliev knot invariant
by
Svetlana D. Tyurina
Pedagogical Institute of Kolomna

S.D.Tyurina¹

e-mail: tyurina.ru

On the universal Vassiliev invariant

Let K be an oriented knot and K^sing_m be a singular knot with m double points. By definition Vassiliev's degree n invariant V_n is invariant vanishing on K^sing_m for for allm > n. We denote by D a chord diagram on the circle and by D_n a linear vector space generated by diagrams with n chords over Q. A linear vector function W_n is called a weight degree n system if it is satisfied 1- and 4-term relations. We consider generic planar projection of a knot with marked "base point". Any crossing x of such projection we're equipped with two "coordinates": \delta_x in {0, 1} and \epsilon_x in { +/- 1}, where \epsilon_x is a local writh number, and \delta_x is defined by the ordering of passing of over- and undercrossings (start at the "base point").

As well known, the Kontsevich integral is the following element of the graded completion \prod_n=0^\inftyD_n of the space D_n:

Z(K) = \infty å m=0   1 (2\pii)m ó õ tmin < t1 < ... < tm < tmax  å P={(zi, z'i)}  (-1)# \downarrow DP m Õ i=1   dzi-dz'i zi-z'i .

The universal Vassiliev knot invariant is defined as the following modified Kontsevich integral: I(K)=[ Z(K)/(Z( \cup )^c(K)/2)], where \cup is the trivial knot, c(K) is a number of critical points of K.

Proposition Formula for computing of the universal Vassiliev invariant modulo 4-degree terms for an arbitrary knot K is the following:

I(K)=1+  1 2 [ å {x, y}  (-1)\deltax+\deltayW2({x, y}) \epsilonx \epsilony[\deltax(1-\deltay)+ \deltay(1-\deltax)] -  1 12 ] Ä +

Picture Omitted

1 4 [ å {x, y, z}  (-1)\deltax+\deltay+\deltazW3({x, y, z}) \epsilonx \epsilony\epsilonz[\deltax(1-\deltay)\deltaz -(1-\deltax)\deltay(1-\deltaz)]],

where sums are taken over all pairs (triplets) of double points of the knot projection.

Bibliography

[1] M.Kontsevich. Vassiliev's knot invariants. - Adv. in Sov.Math. 16, 2, 1993, 137-150.

[2] D.Bar-Natan, S.Garoufalidis, L.Rosansky, D.Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot, preprint, 1997.

[3] J.Lannes, Sur les invariants de Vassiliev de degree inferieur ou egal a 3, L'Enseignement Mathematique, 1993, 39, 295-316.

Footnotes:

¹This work is partly supported by INTAS, grant no. 97-1644.

Date received: February 26, 2000