Evaluating the Jones polynomial with tensor networks.

Abstract

We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot is reduces to the partition function of a q-state anisotropic Potts model with complex interactions, which is defined on a planar signed graph that corresponds to the knot. For any integer q, we cast this partition function into tensor network form, which inherits the interaction graph structure of the Potts model instance, and employ fast tensor network contraction protocols to obtain the exact tensor trace and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace exactly, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.