On 2-variable knot polynomials and Vassiliev invariants

Abstract

Some classical geometric invariants such as crossing number, unknoting number, bridge number and so on are known not to be of finite type. It is known that the coefficients of the Jones polynomial are not finite type invariants while the coefficients of the Conway polynomial are finite type invariants. We show that the nontrivial coefficients of the HOMFLY polynomial and the Kauffman polynomial of a knot are not finite type invariants by constructing examples using the trefoil, figure eight knot and torus knots.