A Geometric Characterization of Vassiliev Invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree ≤ m if and only if it is a polynomial of degree ≤ m on every geometric sequence of knots. Here a sequence Kz with z ∈ Z is called geometric if the knots Kz coincide outside a ball B, inside of which they satisfy Kz ∩B = τz for all z and some pure braid τ . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in S1 × S2 that can be distinguished by Z/2-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over Z a universal Vassiliev invariant of degree 1 for knots in S1×S2. Introduction and statement of results A Vassiliev invariant is a map v : K → A from the set of knots K to an abelian group A such that v satisfies a certain finiteness condition (see §1). Vassiliev invariants are commonly interpreted as polynomials on the set of knots [2, 3, 21]. One instance of this analogy is the following criterion: Theorem (J.Dean [6], R.Trapp [20]). A Vassiliev invariant v : K → Q of degree ≤ m is a polynomial of degree ≤ m on every twist sequence of knots. A twist sequence is a family of knots Kz (indexed by z ∈ Z) that are the same outside a ball, inside of which they differ as depicted in Figure 1. Using this criterion, J.Dean and R.Trapp showed that the class of Vassiliev invariants does not contain certain classical knot invariants such as crossing number, genus, signature, unknotting number, bridge number or braid index. K K K K K -2 -1 0 1 2 ... ... Figure 1. Local picture of a twist sequence of knots Date: June 30, 2003 (revised version). 1991 Mathematics Subject Classification. 57M27, 57M25, 20F36.