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HOMOLOGY FOR QUANDLES WITH PARTIAL GROUP OPERATIONS
Scott Carter1, Atsushi Ishii2, Masahico Saito3
1Department of Mathematics and Statistics, University of South Alabama, ILB 325, Mobile, AL 36688, United States.
Pacific Journal of Mathematics
|April 8, 2021
Summary
This study introduces a unified homology theory for multiple conjugation quandles (MCQs), merging group and quandle homology. This new framework enables the development of cocycle invariants for handlebody-links.
Area of Science:
- Algebraic Topology
- Knot Theory
- Homology Theory
Background:
- Quandles are algebraic structures related to knots and knotted surfaces, with existing homology theories analogous to group homology.
- Previous research applied quandle homology to knot and surface theory, highlighting the need for a more comprehensive approach.
- Multiple Conjugation Quandles (MCQs) are introduced as a specific type of quandle, integrating group structures via conjugation.
Purpose of the Study:
- To define a unified homology theory that integrates both group and quandle homology.
- To develop a novel homology theory specifically for Multiple Conjugation Quandles (MCQs), considering their unique algebraic properties.
- To establish new invariants for handlebody-links using the developed cocycle theory.
Main Methods:
- Definition of a new homology theory for MCQs, incorporating both quandle operations and inter-group compatibilities.
- Characterization of the first homology group for MCQs.
- Introduction of extensions by 2-cocycles and definition of degenerate subcomplexes within simplicial decompositions.
Main Results:
- A unified homology theory for MCQs is successfully established, bridging group and quandle homology.
- The first homology group of MCQs is explicitly characterized.
- Cocycle invariants are defined for handlebody-links, leveraging the new homology theory.
Conclusions:
- The developed homology theory provides a powerful tool for studying MCQs and their applications.
- The new cocycle invariants offer novel methods for distinguishing and analyzing handlebody-links.
- This work unifies algebraic structures and topological invariants, opening avenues for further research in knot theory and related fields.
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