The topological fundamental group and generalized covering spaces

We give a criterion to detect whether the derivatives of knot polynomials at a point are Vassiliev invariants or not. As an application we show that for each nonnegative integer n, JK(n)(a) is a Vassiliev invariant if and only if a=1, where JK(n)(a) is the nth derivative of the Jones polynomial JK(t) of a knot K at t=a. Similarly we apply the criterion for the Conway, Alexander, Q-, HOMFLY and Kauffman polynomial. e give two methods of constructing a polynomial invariant from a numerical Vassiliev invariant of degree n, by using a sequence of knots induced from a double dating tangle. These two new polynomial invariants are Vassiliev invariants of degree ⩽n and their values on a knot are also polynomials of degree ⩽n.