Knot and Link Invariants

In this chapter we make some historical observations and comment on some early work in knot theory. Invariants of knots and links are introduced in Section 11.2. Witten’s interpretation of the Jones polynomial via the Chern–Simons theory is discussed in Section 11.3. A new invariant of 3-manifolds is obtained as a byproduct of this work by an evaluation of a certain partition function of the theory. We already met this invariant, called the Witten–Reshetikhin–Turaev (or WRT) invariant in Chapter 10. In Section 11.4 we discuss the Vassiliev invariants of singular knots. Gauss’s formula for the linking number is the starting point of some more recent work on self-linking invariants of knots by Bott, Taubes, and Cattaneo. We will discuss their work in Section 11.5. The self-linking invariants were obtained earlier by physicists using Chern–Simons perturbation theory. This work now forms a small part of the program initiated by Kontsevich [235] to relate topology of low-dimensional manifolds, homotopical algebras, and non-commutative geometry with topological field theories and Feynman diagrams in physics. See also the book [176] by Guadagnini. Khovanov’s categorification of the Jones polynomial by Khovanov homology is the subject of Section 11.6. We would like to remark that in recent years many applications of knot theory have been made in chemistry and biology (for a brief of discussion of these and further references see, for example, [260]). Some of the material in this chapter is from my article [263]).