Louis H. Kauffman

A Quantum Context for the Jones Polynomial and Khovanov Homology

Abstract

We give a quantum statistical interpretation for the bracket polynomial state sum $\langle K \rangle$, the Jones polynomial $V_K(t)$ and for virtual knot theory versions of the Jones polynomial, including the arrow polynomial. We use these quantum mechanical interpretations to give new quantum algorithms for these polynomials. In those cases where Khovanov homology is defined, the Hilbert space $C(K)$ of our model is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. There is a natural unitary transformation $U: C(K) \to C(K)$ such that $\langle K \rangle = \text{Tr}(U)$ where $\langle K\rangle$ denotes the evaluation of the state sum model for the corresponding polynonmial. We show that for the Khovanov boundary operator $d: C(K) \to C(K)$, we have the relationship $dU + Ud = 0.$ Consequently, the operator $U$ acts on the Khovanov homology, and we obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. We raise the question of the relationship of this model with recent work of Witten.