Biquandle (co)homology and handlebody-links

TL;DR

This paper develops a new (co)homology theory for multiple conjugation biquandles and constructs invariants for handlebody-links using 2-cocycles, extending previous biquandle invariants to more complex topological objects.

Contribution

It introduces the (co)homology group of multiple conjugation biquandles and constructs handlebody-link invariants using 2-cocycles, linking biquandle theory with handlebody-link topology.

Findings

Defined (co)homology groups for multiple conjugation biquandles.

Constructed handlebody-link invariants via 2-cocycles.

Extended biquandle cocycle invariants to handlebody-links.

Abstract

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for $S^1$-oriented handlebody-links using $2$-cocycles. When a multiple conjugation biquandle $X\times\mathbb{Z}_{\operatorname{type}X_Y}$ is obtained from a biquandle $X$ using $n$-parallel operations, we provide a $2$-cocycle (or $3$-cocycle) of the multiple conjugation biquandle $X\times\mathbb{Z}_{\operatorname{type}X_Y}$ from a $2$-cocycle (or $3$-cocycle) of the biquandle $X$ equipped with an $X$-set $Y$.