In this chapter we introduce the main protagonist of this book: the finite type, or Vassiliev knot invariants.

First we define the Vassiliev skein relation and extend, with its help, arbitrary knot invariants to knots with double points. A Vassiliev invariant of order at most n is then defined as a knot invariant which vanishes identically on knots with more than n double points.

After that, we introduce a combinatorial object of great importance: the chord diagrams. Chord diagrams serve as a means to describe the symbols (highest parts) of the Vassiliev invariants.

Then we prove that classical invariant polynomials are all, in a sense, of finite type, explain a simple method of calculating the values of Vassiliev invariants on any given knot and give a table of basis Vassiliev invariants up to degree 5.

Finally, we show how Vassiliev invariants can be defined for framed knots and for arbitrary tangles.

Definition of Vassiliev invariants

The original definition of finite type knot invariants was just an application of the general machinery developed by V. Vassiliev to study complements of discriminants in spaces of maps.

The discriminants in question are subspaces of maps with singularities of some kind. In particular, consider the space of all smooth maps of the circle into ℝ 3. Inside this space, define the discriminant as the subspace formed by maps that fail to be embeddings, such as curves with self-intersections, cusps, etc.