Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial

TL;DR

This paper investigates the homology of specific Yang-Baxter operators related to the HOMFLYPT polynomial, providing explicit formulas for low-dimensional cases and simplifying the computation process.

Contribution

It introduces a reduction method for computing the homology of Yang-Baxter operators and derives explicit formulas for the third and fourth homology groups.

Findings

Reduction of homology computation to initial conditions

Explicit formulas for third and fourth homology groups

Simplified approach to Yang-Baxter operator homology

Abstract

In this paper, we analyze the homology of the Yang-Baxter Operators $R_{(m)}$ yielding the HOMFLYPT polynomial, reducing the computation of the $n$-th homology of $R_{(m)}$ for arbitrary $m$ to the computation of $n+1$ initial conditions. We then produce the explicit formulas for the third and fourth homology.