Homology for quandles with partial group operations

Scott Carter, Atsushi Ishii, Masahico Saito and Kokoro Tanaka

Vol. 287 (2017), No. 1, 19–48

DOI: 10.2140/pjm.2017.287.19

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. The homology theory defined here for MCQs takes 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.

Keywords

quandle, homology, handlebody-link

Mathematical Subject Classification 2010

Primary: 57M15, 57M25, 57M27, 57Q45

Secondary: 55N99, 18G99

Milestones

Received: 8 July 2015

Revised: 15 July 2016

Accepted: 29 July 2016

Published: 6 February 2017