A determinant formula for the Jones polynomial of pretzel knots

TL;DR

This paper introduces a new determinant formula for computing the Jones polynomial of pretzel knots using a weighted adjacency matrix derived from a plane bipartite graph, linking knot invariants with graph theory and algebraic structures.

Contribution

It provides a novel algorithm connecting the Jones polynomial of pretzel knots to perfect matchings and Tutte polynomial activities, bridging knot theory and graph theory.

Findings

Determinant of the weighted adjacency matrix equals the Jones polynomial.

Weights are Tutte's activity letters, connecting to Tutte polynomial.

Evaluation relates to Khovanov homology chain complexes.

Abstract

This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte's activity letters that arise because the Jones polynomial is a specialization of the signed version of the Tutte polynomial. The relationship is formalized between the familiar spanning tree setting for the Tait graph and the perfect matchings of the plane bipartite graph above. Evaluations of these activity words are related to the chain complex for the Champanerkar-Kofman spanning tree model of reduced Khovanov homology.