Carter Scott, Ishii Atsushi, Saito Masahico, Tanaka Kokoro
Department of Mathematics and Statistics, University of South Alabama, ILB 325, Mobile, AL 36688, United States.
Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Ibaraki, Tsukuba 305-8571, Japan.
Pac J Math. 2017;287(1):19-48. doi: 10.2140/pjm.2017.287.19. Epub 2017 Feb 6.
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.
quandles是一种具有二元运算的集合，该运算满足与赖德迈斯特移动相对应的三个条件。quandles的同调理论是以类似于群同调的方式发展起来的，并已应用于纽结和打结曲面。本文定义了一种统一群同调理论和quandle同调理论的同调理论。一个由群的并集组成且运算在每个群分量上限制为共轭的quandle被称为多重共轭quandle（MCQ，在内部有严格定义）。在此定义中，施加了群运算和quandle运算之间的相容性，这是由对柄体链环着色的考虑所激发的。这里为MCQ定义的同调理论同时考虑了群运算和quandle运算及其相容性。刻画了第一同调群，并给出了由2 - 上闭链扩张的概念。退化子复形是相对于棱柱形（单纯形的乘积）复形的单纯分解和群逆来定义的。还为柄体链环定义了上闭链不变量。
