Polynomial Link Invariants from Markov Traces on Braid Group Representations

Abstract:

In 1984 Vaughan Jones discovered a polynomial knot invariant that is now called the Jones polynomial. This talk will explore the process of obtaining polynomial knot invariants from Markov traces on braid group representations. In particular, we will focus on the Jones polynomial, and we will see two different constructions of the polynomial as a Markov trace on the Temperley-Lieb algebra. We will explore Jones' original algebraic construction as well as a diagrammatic construction due to the work of Kauffman. Finally, we will briefly address two generalisations of the Jones polynomial: the HOMFLY polynomial which is a Markov trace on the Hecke algebra and the Kauffman polynomial which is a Markov trace on the BMW algebra.