A determinant formula of the Jones polynomial for a family of braids

TL;DR

This paper extends a determinant-based method to compute the Jones polynomial for specific braid subfamilies and calculates the Kauffman polynomial for (2,q) torus knots efficiently, broadening computational techniques for quantum invariants.

Contribution

It generalizes Cohen et al.'s determinant approach to new braid subfamilies and introduces polynomial-time computation of the Kauffman polynomial for (2,q) torus knots.

Findings

Jones polynomial for certain braids can be computed as a determinant.

Kauffman polynomial for (2,q) torus knots computed in polynomial time.

Method extends to quantum invariants beyond HOMFLYPT polynomial.

Abstract

In 2012, Cohen, Dasbach, and Russell presented an algorithm to construct a weighted adjacency matrix for a given knot diagram. In the case of pretzel knots, it is shown that after evaluation, the determinant of the matrix recovers the Jones polynomial. Although the Jones polynomial is known to be #P-hard by Jaeger, Vertigan, and Welsh, this presents a class of knots for which the Jones polynomial can be computed in polynomial time by using the determinant. In this paper, we extend these results by recovering the Jones polynomial as the determinant of a weighted adjacency matrix for certain subfamilies of the braid group. Lastly, we compute the Kauffman polynomial of (2,q) torus knots in polynomial time using the balanced overlaid Tait graphs. This is the first known example of generalizing the methodology of Cohen to a class of quantum invariants which cannot be derived from the HOMFLYPT polynomial.