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Computing the associated groups of quandles with tools from group homology theory

Katsumi Ishikawa

Abstract

We give a complete description of the associated group of any quandle as a central extension of the inner-automorphism group. As an application, we compute the second quandle homology groups of quandles of some families, including those of Alexander quandles.

Computing the associated groups of quandles with tools from group homology theory

Abstract

We give a complete description of the associated group of any quandle as a central extension of the inner-automorphism group. As an application, we compute the second quandle homology groups of quandles of some families, including those of Alexander quandles.
Paper Structure (9 sections, 11 theorems, 40 equations)

This paper contains 9 sections, 11 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Associated groups of quandles
  4. Proof of Theorem \ref{['main-thm']}
  5. Preparation for applications
  6. Applications
  7. Alexander quandles
  8. Connected generalized Alexander quandles
  9. Products of connected quandles

Key Result

Theorem 1.1

Let $M$ be a $\mathbb{Z}[T^{\pm 1}]$-module and $X = Q_{M,T}$ the Alexander quandle. Then, we have where $T$ acts on $\Lambda^2 (1-T)M$ diagonally.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: eis2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['main-thm']}
  • Lemma 3.3
  • proof
  • ...and 10 more