Approximation of knot invariants by Vassiliev invariants

Abstract

We give a criterion for a knot invariant, which is additive under connected sum, to be approximated by a sequence of Vassiliev (finite-type) invariants. This partially answers the question: can an arbitrary knot invariant be approximated by Vassiliev invariants? A knot invariant, which is additive under connected sum, is approximated by a sequence of Vassiliev invariants if and only if for any knot K it is constant on the infinite intersection ${\cap}K(n)$. Here K(n) is the set of knots whose class in the group GK\rm\sb{n} of the classes of n-equivalent knots (due to Gusarov) differs from the class of K by some torsion element of GK\rm\sb{n}. Roughly speaking, ${\cap}K(n)$ is the set of knots which cannot be distinguished from K by any Vassiliev invariant. Thus, it is impossible to solve the problem of approximating a knot invariant by Vassiliev invariants without answering the question which knots can be separated by Vassiliev invariants. It is also shown that generalized signatures and certain Minkowski units are not Vassiliev invariants