A permanent formula for the Jones polynomial

TL;DR

This paper establishes a novel connection between the Jones polynomial of a link and the permanent of a matrix, enabling quantum algorithms to efficiently approximate the polynomial for decision problems.

Contribution

It introduces a method to compute the Jones polynomial via the permanent of a specially constructed matrix, linking knot invariants with quantum computational complexity.

Findings

The permanent of a 7n by 7n matrix equals the Jones polynomial of a link.

This connection enables Monte-Carlo algorithms for approximating the Jones polynomial.

The approach relates to quantum algorithms for problems in BQP.

Abstract

The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with $n$ crossings, we define a 7n by 7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time using quantum resources.