ABSTRACT
This chapter focuses on the theory of Vassiliev invariants and to some of its numerous connections with other parts of mathematics and mathematical physics. It also focuses on only one theorem, the basic existence theorem for invariants with a given “mth derivative”. Any invariant V of oriented knots in oriented space can be extended to an invariant of singular knots. It is easy to show that many known knot invariants are Vassiliev, including, for example, all coefficients of the Conway, Jones, and Homfly polynomials. The chapter shows that Z(K) is also invariant under deformations of K that do move critical points. The idea is to narrow the parts near critical points to sharp needles using horizontal deformations, and then show that very sharp needles contribute almost nothing to Z(K) and therefore can be moved around freely.
