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Quandles with one non-trivial column

Nicholas Cazet

Abstract

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and $Hom$ quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.

Quandles with one non-trivial column

Abstract

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.
Paper Structure (12 sections, 21 theorems, 31 equations, 4 figures, 8 tables)

This paper contains 12 sections, 21 theorems, 31 equations, 4 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Definitions
  3. Automorphism Groups
  4. Quandle Polynomials
  5. $Hom$ Quandles
  6. Quandle Coloring Number
  7. Quandle Quivers
  8. Quandle Cohomology
  9. Quandle Cocycle Invariant
  10. The quandle $P_n^\sigma$
  11. Cohomology groups of $P_n^\sigma$
  12. The quiver and cocycle invariant of links using $P_n^\sigma$

Key Result

Theorem 2.8

Let $X$ and $A$ be quandles. If $A$ is abelian, then $Hom(X,A)$ is an abelian quandle under the pointwise operation $(f*g)(x)=f(x)*g(x)$. Let $X$ be a finitely generated quandle and $A$ an abelian quandle. Then $Hom(X,A)$ is isomorphic to a subquandle of $A^c$ where $c$ is the minimal number of gene The pointwise operation on $Hom(X,A)$ agrees with the componentwise operation on the product $A^c$.

Figures (4)

  • Figure 1: Quandle coloring condition.
  • Figure 2: Crossing types between 2 components.
  • Figure 3: Removing the interiors of disk neighborhoods.
  • Figure 4: Replacing an edge with a twist tangle.

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2: symquandle
  • Example 2.3: symquandle
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8: Crans, Nelson '14 crans2014hom
  • Definition 2.9
  • Theorem 2.10
  • ...and 40 more