Some computational results on mod 2 finite-type invariants of knots and string links

Abstract

In [11], Vassiliev described a new way to obtain invariants of knots in S3 . His paper contains the outline of an algorithm for computing his invariants. Gusarov [5] obtained the same set of invariants independently and by different methods. These invariants of Vassiliev and Gusarov are now often referred to as finite-type invariants. At the end of this paper there are tables of primitive invariants of order ≤ 6 for knots of ≤ 10 crossings. These invariants were computed using an implementation of Vassiliev’s algorithm, which is described in [9] . In order to create tables such as these, a basis must be chosen for the invariants of order ≤ 6. In Section 2 we will discuss the choice of such a basis, and make some related observations. In Section 3 we note that the algorithm for computing knot invariants extends easily to the computation of finite-type invariants of string links. We describe a computation which shows that there is a mod-2 weight system of order 5 (first noted by Kneissler and Dogolazky) for 2-strand string links which does not “integrate” to a mod-2 finite-type invariant of order 5. In Section 4 we present the two matrices for translating our numbers into finitetype invariants obtained from the derivatives of knot polynomials, following the notation of Kanenobu [7].