Evaluating the Jones polynomial with tensor networks

TL;DR

This paper presents tensor network algorithms for efficiently evaluating the Jones polynomial of complex knots, enabling calculations that surpass previous methods in scalability and speed.

Contribution

It introduces a tensor network contraction approach to compute the Jones polynomial for arbitrary knots, demonstrating subexponential scaling in typical cases.

Findings

Tensor network contraction algorithms can evaluate the Jones polynomial.

The method scales subexponentially with the number of crossings.

It enables analysis of more complex knots than traditional methods.

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 maps to the partition function of a $q$-state Potts model defined as a planar graph with weighted edges that corresponds to the knot. For any integer $q$, we cast this partition function into tensor network form 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, 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.