On the Jones polynomial of 2n-plat presentations of knots

TL;DR

This paper introduces a matrix-based method to compute the Jones polynomial for 2n-plat knot presentations, generalizing previous techniques for 6-plat knots, thus providing a systematic approach for these calculations.

Contribution

It develops a generalized method using 5x5 matrix representations of the braid group to calculate Jones polynomials for any 2n-plat knot presentations.

Findings

Provides a systematic matrix-based calculation method

Generalizes previous 6-plat techniques to 2n-plat knots

Enables computation of Jones polynomial for broader knot classes

Abstract

In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group $\mathbb{B}_6$ into a group of $5\times 5$ matrices. We also can calculate the Jones polynomial of the $2n$-plat presentations of knots by generalizing the method for the 6-plat presentations of knots.