Framed knots and cabling operations

In this chapter we show how to associate to a framed knot K an infinite set of framed knots and links, called the (p, q)-cables of K. The operations of taking the (p, q)-cable respect the Vassiliev filtration, and give rise to operations on Vassiliev invariants and on chord diagrams.We shall give explicit formulae that describe how the Kontsevich integral of a framed knot changes under the cabling operations. As a corollary, this will give an expression for the Kontsevich integral of all torus knots.Framed version of the Kontsevich integralIn order to describe a framed knot, one only needs to specify the corresponding unframed knot and the self-linking number. This suggests that there should be a simple formula to define the universal Vassiliev invariant for a framed knot via the Kontsevich integral of the corresponding unframed knot. This is, indeed, the case, as we shall see in Section 9.1.2. However, for our purposes it will be more convenient to use a definition of the framed Kontsevich integral given by V. Goryunov (1999) which is in the spirit of the original formula of Kontsevich described in Section 8.2.Remark 9.1. For framed knots and links, the universal Vassiliev invariant was first defined by Le and Murakami (1996a), who gave a combinatorial construction of it using the Drinfeld associator (see Chapter 10). Goryunov used his framed Kontsevich integral in (Goryunov 1998) to study Arnold's J+-theory of plane curves (or, equivalently, Legendrian knots in a solid torus).