Set-theoretic Yang-Baxter cohomology of cyclic biquandles

TL;DR

This paper fully computes the free and torsion parts of the set-theoretic Yang-Baxter cohomology groups for finite cyclic biquandles, advancing understanding of their algebraic structure and partially confirming a conjecture about their homology.

Contribution

It provides complete calculations of the free and torsion components of these cohomology groups and offers bounds for higher-dimensional torsions, addressing a key conjecture.

Findings

Determined free parts of the cohomology groups.

Computed torsion subgroups of the 1st and 2nd homology groups.

Provided bounds for torsion in higher homology groups.

Abstract

We completely determine the free parts of the set-theoretic Yang-Baxter (co)homology groups of finite cyclic biquandles, along with fully computing the torsion subgroups of their 1st and 2nd homology groups. Furthermore, we provide upper bounds for the orders of torsions in the 3rd and higher dimensional homology groups. This work partially solves the conjecture that the normalized set-theoretic Yang-Baxter homology of cyclic biquandles satisfy $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}} \oplus \mathbb{Z}_{m}$ when $n$ is odd and $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}}$ when $n$ is even. In addition, we obtain cocycle representatives of a basis for the rational cohomology group of a cyclic biquandle and introduce several non-trivial torsion homology classes.