Biquandle (co)homology and 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 $X\times {\mathbb{Z}}_{\mathrm{type}{X}_Y}$ is obtained from a biquandle $X$ using $n$-parallel operations, we provide a $2$-cocycle (or $3$-cocycle) of the multiple conjugation biquandle $X\times {\mathbb{Z}}_{\mathrm{type}{X}_Y}$ from a $2$-cocycle (or $3$-cocycle) of the biquandle $X$ equipped with an $X$-set $Y$.

arXiv admin note: text overlap with arXiv:1702.01363; text overlap with arXiv:1704.07792 by other authors