# Hecke algebra trace algorithm and some conjectures on weaving knots

## TL;DR

This paper introduces an algorithm for computing trace-based polynomial invariants of knots from braid representations, specifically applied to weaving knots, and explores their topological and geometric properties.

## Contribution

It presents a new algorithm for calculating Hecke algebra traces and polynomial invariants for weaving knots, along with a Mathematica implementation and analysis of their properties.

## Key findings

- Successfully computed polynomial invariants for weaving knots
- Established relationships between topological and geometric invariants
- Generated data for specific weaving knot families

## Abstract

Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In this paper, we provide an algorithm to compute the trace of the Hecke algebra representation of any braid. We simplify this algorithm and write a Mathematica program to compute the invariants such as Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology of a very special family of knots and links $W(n,m)$ known as weaving knots by expressing them as closure of weaving braids. We also explore on the relationship between the topological and geometric invariants of this family of alternating and hyperbolic knots (links) by generating data for the subfamilies $W(3,m)$, $W(4,m)$, $W(5,m)$ and $W(6,m)$ of weaving knots.