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Pointed Quandle Coloring Quivers of Linkoids

Jose Ceniceros, Max Klivans

TL;DR

This work extends quandle coloring techniques from classical knots to linkoids by employing pointed quandles, enabling the construction of pointed quandle coloring quivers and two new linkoid invariants: the in-degree quiver polynomial and the in-degree quiver polynomial matrix. These invariants refine the existing pointed quandle counting invariant and quandle counting matrix by encoding how endomorphisms act on colorings. The authors establish the invariance of these invariants under linkoid moves, and they analyze the pointed quandle coloring quivers for $\mathcal{T}(p,2)$-type linkoids, deriving explicit results depending on gcd conditions and basepoints, including concrete computations for $\widetilde{\mathcal{T}(p,2)}$ and a detailed example with $\widetilde{\mathcal{T}(10,2)}$ over $\mathbb{Z}_5$. The approach yields stronger, computable invariants that distinguish linkoids where traditional invariants fail, with potential impact on the study of knotoids and related entanglement models.

Abstract

We enhance the pointed quandle counting invariant of linkoids through the use of quivers analogously to quandle coloring quivers. This allows us to generalize the in-degree polynomial invariant of links to linkoids. Additionally, we introduce a new linkoid invariant, which we call the in-degree quiver polynomial matrix. Lastly, we study the pointed quandle coloring quivers of linkoids of $(p,2)$-torus type with respect to pointed dihedral quandles.

Pointed Quandle Coloring Quivers of Linkoids

TL;DR

This work extends quandle coloring techniques from classical knots to linkoids by employing pointed quandles, enabling the construction of pointed quandle coloring quivers and two new linkoid invariants: the in-degree quiver polynomial and the in-degree quiver polynomial matrix. These invariants refine the existing pointed quandle counting invariant and quandle counting matrix by encoding how endomorphisms act on colorings. The authors establish the invariance of these invariants under linkoid moves, and they analyze the pointed quandle coloring quivers for -type linkoids, deriving explicit results depending on gcd conditions and basepoints, including concrete computations for and a detailed example with over . The approach yields stronger, computable invariants that distinguish linkoids where traditional invariants fail, with potential impact on the study of knotoids and related entanglement models.

Abstract

We enhance the pointed quandle counting invariant of linkoids through the use of quivers analogously to quandle coloring quivers. This allows us to generalize the in-degree polynomial invariant of links to linkoids. Additionally, we introduce a new linkoid invariant, which we call the in-degree quiver polynomial matrix. Lastly, we study the pointed quandle coloring quivers of linkoids of -torus type with respect to pointed dihedral quandles.
Paper Structure (6 sections, 17 theorems, 49 equations, 11 figures)

This paper contains 6 sections, 17 theorems, 49 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Linkoids
  3. Quandles and Pointed Quandles
  4. Pointed Quandle Coloring Quivers
  5. The In-degree Quiver Polynomial and Matrix
  6. Pointed Quandle Coloring Quivers of Linkoids of $\mathcal{T}(p,2)$-type

Key Result

Theorem 4.3

Let $\mathcal{X}$ be a finite $2n$-pointed quandle, $S\subseteq \operatorname{End}(\mathcal{X})$ and $L$ is a oriented $n$-linkoid. Then the quiver $\mathcal{Q}_\mathcal{X}^S(L)$ is an invariant of $L$.

Figures (11)

  • Figure 1: A minimal generating set of oriented Reidemeister moves.
  • Figure 2: The S-move.
  • Figure 3: The over and under forbidden moves.
  • Figure 4: A 1-linkoid diagram of $\mathcal{T}(4,2)$-type
  • Figure 5: Quandle relations at a positive and negative crossing.
  • ...and 6 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 44 more