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Quandle Coloring Quivers of general Torus links by dihedral quandles

Mohamed Elhamdadi, Brooke Jones, Minghui Liu

Abstract

We completely characterize the coloring quivers of general torus links by dihedral quandles by first exhausting all possible numbers of colorings, followed by determining the interconnections between colorings in each case. The quiver is obtained as function of the number of colorings. The quiver always contains complete subgraphs, in particular a complete subgraph corresponding to the trivial colorings, but the total number of subgraphs in the quiver and the weights of their edges varies depending on the number of colorings.

Quandle Coloring Quivers of general Torus links by dihedral quandles

Abstract

We completely characterize the coloring quivers of general torus links by dihedral quandles by first exhausting all possible numbers of colorings, followed by determining the interconnections between colorings in each case. The quiver is obtained as function of the number of colorings. The quiver always contains complete subgraphs, in particular a complete subgraph corresponding to the trivial colorings, but the total number of subgraphs in the quiver and the weights of their edges varies depending on the number of colorings.
Paper Structure (4 sections, 12 theorems, 32 equations, 7 figures, 4 tables)

This paper contains 4 sections, 12 theorems, 32 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A review of Quandles and link colorings
  3. Quandle Coloring Quivers of $(5,q)$ and $(7,q)$-Torus links with Dihedral Quandles
  4. Quandle Coloring Quivers of General $(p,q)$-Torus links with Dihedral Quandles

Key Result

Theorem 3.1

If $|Hom(T(5,q), R_n)|=n$, then the full quandle coloring quiver is the complete directed graph:

Figures (7)

  • Figure 1: Colorings of arcs at positive and negative crossings
  • Figure 2: Oriented figure-8 knot with labeled arcs
  • Figure 3: Coloring of the braid $(\sigma_1 \sigma_2 \sigma_3 \sigma_4)^q$ whose closure is $T(5,q)$
  • Figure 4: $(\overleftrightarrow{K_5}, \hat{5})$
  • Figure 5: $(\overleftrightarrow{K_{20}}, \hat{20})$
  • ...and 2 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Example 3.5
  • ...and 18 more