Diagram formulas of Viro-Polyak type and the Kontsevich integral for (2, n)-torus knots
by
Svetlana D. Tyurina
Kolomna Pedagogical Institute

It is well known that universal Vassiliev knot invariant is the most strong invariant among any ones. It is computed with the help of the Kontsevich integral which gives a series expansion of Vassiliev invariants of knots. Every term of the Kontsevich series is a chord diagram with numerical coefficient. Finite type invariants are determined by weights of chord diagrams.

In 1997 D.Bar-Natan obtained the formula of universal Vassiliev invariant for trivial knot.

We discuss a formula for computing no more 4-order terms of the Kontsevich integral for (2, n)-type torus knots.

We use the approuch of T.Q.T. Le and J. Murakami (¹), (²) to obtain the formula for the universal Vassiliev invariant of torus knots.

Theorem. The formula for computing the Kontsevich integral I for (2, n)-type torus knots is following:

I(2, n) = ~ \Phi   -2   (\Omega12, \Omega23) ·\Phi(\Omega12, -\Omega23)  cdot P ·\Phi(-\Omega23, \Omega12),

where     P=\sum_m=0^\infty(\fracn2)^m·\fracD_mm! = exp(\frac12
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) ·
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,        D_m =
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\Phi(\Omega_ij, \Omega_kl) is Drinfeld's associator,

\Omega_ij is a chord diagram on the trivial 3-strands tangle with a chord connected i-th and j-th strands,

[(\Phi)\tilde] is obtained from \Phi by closing each tangle to \cup -knot.

The formula for computing no more 4-order terms of the Kontsevich integral I for (2, n)-type torus knots is following:
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I(2, n) = 1 + \frac3n2 - 424 Ä + \fracn3 - n48+\frac15n4 - 165760(- 8).

We analyse this formula with respect to next diagram formulas of Viro-Polyak type:

V₂(2, n)= <
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, G > ,

V₃(2, n) = <
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, G > ,

V₄(2, n) = \frac12 <
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+
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, G >

and formulas of following type:
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V2(2, n) = \frac3n2 - 324 Ä ;        V3(2, n) = \fracn3 - n48;        ...

Footnotes:

¹ T.Q.T. LE and J. MURAKAMI, Representations of the category of tangles by Kontsevich's iterated integral, Comm. Math. Phys. 168 (1995), 535-562.

² T.Q.T. LE and J. MURAKAMI, Kontsevich's integral for the Kauffman polynomial, Nagoya Math. J. Vol.142 (1996), 39-65.

Date received: June 7, 1999