Vassiliev Invariants as Polynomials

Three results are shown which demonstrate how Vassiliev invariants behave like polynomials. 0. Introduction. The purpose of this note is to show how the phrase \Vassiliev invariants are like polynomials" can be a useful working paradigm. This has been illustrated in other talks at this conference: Deguchi used Vassiliev invariants because they are computable in polynomial time (see below) and Burri 4] demonstrated that, for a xed shadow, Vassiliev invariants are polynomial functions in the gleams. Here three other results are presented which reeect the motto in some manner. They are as follows: the fact that Vassiliev invariants form a ltered algebra; Dean and Trapp's criterion for an invariant to be of nite type; and a partial solution to the problem of integrating a weight system. Two results with a polynomial avour which don't quite t into the framework of this note are that the value of an invariant of degree n on a knot with c crossings is bounded by a polynomial of degree n in c and that the value is calculable in similarly polynomial time. These follow from Stanford's algorithm 11] for calculating Vassiliev invariants and were also proved by Bar-Natan 2]. For simplicity all invariants will take values in Q.