Integrating a weight system of order n to an invariant of (n−1)-singular knots

Abstract

Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.

On the First Two Vassiliev Invariants

“…The first of these bounds was obtained by Lin and Wang [10] and led Bar-Natan [2] to prove that any type n invariant is bounded by a degree n polynomial in the crossing number -this also follows from Stanford's algorithm [16] for calculating Vassiliev invariants. The bound for v 3 was obtained in [19] by utilizing Domergue and Donato's integration [6] of a type three weight system.…”