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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
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PubMed
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.

Keywords:
handlebody-linkhomologyquandle

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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.