Knots and distributive homology: from arc colorings to Yang-Baxter homology

TL;DR

This paper explores the connection between distributive homology, arc colorings, and Yang-Baxter operators, proposing a framework that links knot invariants with algebraic structures like biquandles and potential categorification.

Contribution

It introduces a novel approach to Yang-Baxter homology, extending the understanding of algebraic invariants in knot theory and relating them to distributive homology and biquandles.

Findings

Defined and visualized Yang-Baxter homology

Provided a simple description of biquandle homology

Surveyed the relation between arc colorings and algebraic structures

Abstract

This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology, in particular giving a simple description of homology of biquandles.