The Khovanov Homology of Rational Tangles

Abstract

The aim of this thesis is to describe the Khovanov homology of rational tangles. To this extent we describe rational tangles, followed by Khovanov homology, then combine the two at the end. In Chapter 1 we review the main points of the theory of rational tangles. In particular, we show that rational tangles are classified by a function known as the tangle fraction, which associates to each rational tangle a rational number. This classification theorem implies that each rational tangle can be constructed by adding and multiplying together multiple copies of certain types of tangle. In Chapter 2 we review the Khovanov homology theory we will use to study rational tangles. After discussing link invariants, the Kauffman bracket, and categorification in general, we develop a generalization of Khovanov homology to tangles due to Bar-Natan [Bar04]. We then specialize this to a ‘dotted’ theory, which is equivalent to Khovanov’s original theory [Kho99] on links and is significantly easier to work with. In Chapter 3, we combine the previous topics to develop a theory of the Khovanov homology of rational tangles. We determine the Khovanov complexes of integer tangles, before presenting a helpful isomorphism and discussing its implications. We then combine these to prove Theorem 3.3.1, the main result of this thesis. Finally, in Chapter 4 we briefly discuss some of the application of Theorem 3.3.1. Unexpectedly, we find that the bigradings of the subobjects in the Khovanov complexes can be described by matrix actions, and that one can recover from this action the reduced Burau representation of B3, the three-strand braid group. The full ramifications of this observation are yet to be determined!