Vol. 287, No. 1, 2017

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ISSN: 0030-8730
Homology for quandles with partial group operations

Scott Carter, Atsushi Ishii, Masahico Saito and Kokoro Tanaka

Vol. 287 (2017), No. 1, 19–48
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
Authors
Scott Carter
Department of Mathematics and Statistics
University of South Alabama
ILB 325
Mobile, AL 36688
United States
Atsushi Ishii
Institute of Mathematics
University of Tsukuba
1-1-1 Tennodai
Ibaraki
Tsukuba 305-8571
Japan
Masahico Saito
Department of Mathematics and Statistics
University of South Florida
4202 E Fowler Ave.
PHY 114
Tampa, FL 33620
United States
Kokoro Tanaka
Department of Mathematics
Tokyo Gakugei University
4-1-1 Nukuikita-machi
Koganei-shi
Tokyo 184-8501
Japan