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Set-theoretic Yang-Baxter cohomology of cyclic biquandles

Minyi Liang, Xiao Wang, Seung Yeop Yang

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.

Set-theoretic Yang-Baxter cohomology of cyclic biquandles

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 when is odd and when 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.

Paper Structure

This paper contains 7 sections, 14 theorems, 39 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cohomology of finite cyclic biquandles
  4. The Betti number of a finite cyclic biquandle
  5. The torsion of a finite cyclic biquandle
  6. Future research
  7. Acknowledgements

Key Result

Proposition 3.2

Suppose that a biquandle $X$ satisfies the following property: Then the coboundary homomorphism $\delta^{n}:C^{n}_{YB}(X;A) \rightarrow C^{n+1}_{YB}(X;A)$ is a $G_{X}$-module homomorphism.

Figures (3)

  • Figure 2.1: An illustration of the set-theoretic Yang–Baxter equation
  • Figure 2.2: A diagrammatic description of the face maps $d_{i,n+1}^{l}$ and $d_{i,n+1}^{r}$
  • Figure 3.1: $\delta^{n}(f \cdot y) = \delta^{n}(f) \cdot y$

Theorems & Definitions (35)

  • Conjecture 1.1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • ...and 25 more