## Invariants of plane curves and Polyak-Viro type formulas for Vassiliev invariants

J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 2, Page 155--175.

Ochiai, Tomoshiro
Invariants of plane curves and Polyak-Viro type formulas for Vassiliev invariants
The Kontsevich integral is decomposed into two parts; one part depends on overpass or underpass of the crossing of a knot while the other depends only on the plane curve obtained by projecting the knot to the plane. In this paper, firstly, we express the latter part in terms of Arnold's invariants of plane curves $J_+$, $J_-$ and $St$ up to degree three. Secondly, we show that the Gauss diagram formulas for the Kontsevich integral agree with other types of formulas for Vassiliev invariants which are introduced by M. Polyak and O. Viro.