Remarks on Khovanov Homology and the Potts Model

TL;DR

This paper explores the connections between Khovanov homology and the Potts model, reformulating statistical mechanics models in topological and quantum terms, and introduces a new quantum algorithm for the Jones polynomial.

Contribution

It provides a self-contained introduction to Khovanov homology, relates it to the Potts model at imaginary temperature, and develops a novel quantum algorithm for computing the Jones polynomial.

Findings

Homological Euler characteristics relate to Potts model partition functions at imaginary temperature.

Reformulation of the Potts model as a quantum amplitude via Wick rotation.

A new quantum algorithm for the Jones polynomial on the unit circle in the complex plane.

Abstract

This paper is dedicated to Oleg Viro on his 60-th birthday. The paper is about Khovanov homology and its relationships with statistical mechanics models such as the Ising model and the Potts model. We give a relatively self-contained introduction to Khovanov homology, and also a reformulation of the Potts model in terms of a bracket state sum expansion on a knot diagram K(G) related to a planar graph G via the medial construction. We consider the original Khovanov homology and also the homology defined by Stosic via the dichromatic polynomial, and examine those values of the Potts model where the partition function can be expressed in terms of homological Euler characteristics. These points occur at imaginary temperature. The last part of the paper reformulates, via a Wick rotation, the Potts model at imaginary temperature as a quantum amplitude and uses the same formalism to give a new quantum algorithm for the Jones polynomial for values of the variable that lie on the unit circle in the complex plane.