Homology for Quandles with Partial Group Operations

TL;DR

This paper introduces a unified homology theory for quandles and groups, focusing on multiple conjugation quandles (MCQs), with applications to knot theory and handlebody-link invariants.

Contribution

It defines a new homology theory for MCQs that combines group and quandle homologies, including characterizations of the first homology group and cocycle invariants.

Findings

Unified homology theory for MCQs developed

Characterization of the first homology group provided

Cocycle invariants for handlebody-links introduced

Abstract

A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group and quandle homology theories. A quandle that is a union of groups with the operation restricting to conjugation on each group component is called a multiple conjugation quandle (MCQ, defined rigorously within). In this definition, compatibilities between the group and quandle operations are imposed which are motivated by considerations on colorings of handlebody-links. A homology theory defined here for MCQs take into consideration both group and quandle operations, as well as their compatibility. The first homology group is characterized, and the notion of extensions by $2$-cocycles is provided. Degenerate subcomplexes are defined in relation to simplicial decompositions of prismatic (products of simplices) complexes and group inverses. Cocycle invariants are also defined for handlebody-links.