Augmented Biracks and their Homology

TL;DR

This paper introduces augmented biracks and their homology theory, extending Yang-Baxter homology, and develops new invariants for classical and virtual knots and links.

Contribution

It defines augmented biracks, formulates their homology, and introduces augmented birack 2-cocycle invariants for knots and links, expanding the algebraic tools in knot theory.

Findings

Extended Yang-Baxter homology to augmented biracks

Defined augmented birack 2-cocycle invariants for knots

Provided examples demonstrating the invariants' effectiveness

Abstract

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang-Baxter homology with a combinatorial formulation for the boundary map and specializes to $N$-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples.