Gauss Diagram Formulas for Vassiliev Invariants

Abstract

Although the original definitions of Vassiliev knot invariants are rather far from explicit formulas, recently several types of formulas for them have been found; see [1], [2], [7], [4], [5]. However, most of them were designed more for the sake of general theory than for actual computations. Our initial goal was to provide more practical formulas. We were motivated by the well-known case of the linking number. It is the simplest of Vassiliev invariants for links. It can be computed in many different ways; see, e.g., [8]. Integral formulas of Bar-Natan [2] and Kontsevich [1] for Vassiliev invariants generalize the Gauss integral formula for the linking number. As is known, the Gauss integral formula has simple combinatorial counterparts. In this paper we present a similar transition to combinatorial formulas for higher-order Vassiliev invariants. As in the case of the linking number, both integral and combinatorial formulas may be obtained from an interpretation of the invariants as degrees of some maps. It was this viewpoint that motivated the whole of our investigations and appeared to be a rich source of various special formulas. We plan to discuss this phenomenon in detail in a forthcoming paper.