Equivalence of two definitions of set-theoretic Yang-Baxter homology

TL;DR

This paper proves the equivalence between two different set-theoretic Yang-Baxter homology definitions, clarifying their relationship and providing examples, including cases with torsion related to the Jones polynomial.

Contribution

It establishes the equivalence of algebraic and graphic homology theories for set-theoretic Yang-Baxter operators, expanding understanding of their structures.

Findings

Proved the equivalence of two homology definitions.

Provided examples of one-term and two-term homologies.

Discovered torsion in homology related to the Jones polynomial.

Abstract

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators(we will call it the "algebraic" version in this article). In 2012, Przytycki defined another homology theory for pre-Yang-Baxter operators which has a nice graphic visualization(we will call it the "graphic" version in this article). We show that they are equivalent. The "graphic" homology is also defined for pre-Yang-Baxter operators, and we give some examples of it's one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang-Baxter operator that yields the Jones polynomial.