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Classifying links and spatial graphs with finite $N$-quandles

Blake Mellor

TL;DR

This work develops a unified framework linking fundamental quandles and their $N$-quotients to cosets of peripheral subgroups, extending the theory from knots and links to spatial graphs. It proves that finite $N$-quandles correspond to finite orbifold fundamental groups $\pi^N_1$, enabling a complete classification of links with finite $N$-quandles and providing a spherical-orbifold interpretation that connects combinatorial invariants with 3-dimensional geometry. The main contributions include establishing isomorphisms between $Q_N(\Gamma)$ and coset-based quandles, proving finiteness results for $Conj_N(Q)$, and giving explicit classifications and sizes (with Appendix tables) for links and certain spatial graphs. The results bridge knot theory, quandle theory, and orbifold geometry, offering practical criteria for finiteness and enabling comprehensive enumeration in many cases.

Abstract

The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the peripheral subgroup(s) in the fundamental group of the knot or link. We extend these relationships to spatial graphs, and to $N$-quandles of links and spatial graphs. As an application, we are able to give a complete list of links with finite $N$-quandles, proving a conjecture from \cite{MS}, and a partial list of spatial graphs with finite $N$-quandles.

Classifying links and spatial graphs with finite $N$-quandles

TL;DR

This work develops a unified framework linking fundamental quandles and their -quotients to cosets of peripheral subgroups, extending the theory from knots and links to spatial graphs. It proves that finite -quandles correspond to finite orbifold fundamental groups , enabling a complete classification of links with finite -quandles and providing a spherical-orbifold interpretation that connects combinatorial invariants with 3-dimensional geometry. The main contributions include establishing isomorphisms between and coset-based quandles, proving finiteness results for , and giving explicit classifications and sizes (with Appendix tables) for links and certain spatial graphs. The results bridge knot theory, quandle theory, and orbifold geometry, offering practical criteria for finiteness and enabling comprehensive enumeration in many cases.

Abstract

The fundamental quandle is a complete invariant for unoriented tame knots \cite{JO, Ma} and non-split links \cite{FR}. The proof involves proving a relationship between the components of the fundamental quandle and the cosets of the peripheral subgroup(s) in the fundamental group of the knot or link. We extend these relationships to spatial graphs, and to -quandles of links and spatial graphs. As an application, we are able to give a complete list of links with finite -quandles, proving a conjecture from \cite{MS}, and a partial list of spatial graphs with finite -quandles.
Paper Structure (9 sections, 11 theorems, 13 equations, 14 figures, 3 tables)

This paper contains 9 sections, 11 theorems, 13 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Quandles and $N$-quandles
  3. Definitions and notation
  4. Fundamental quandles of Links and Spatial Graphs
  5. Wirtinger presentations for fundamental quandles
  6. A topological interpretation of the fundamental quandle
  7. Relating $Q_N(\Gamma)$ to cosets in $\pi_1(\mathbb{S}^3 - \Gamma)/\langle \mu_i^{n_i} \rangle$
  8. Links and spatial graphs with finite $N$-quandles
  9. Questions for further investigation

Key Result

Lemma 2.1

If $x, y, u$ and $v$ are elements of a quandle, then

Figures (14)

  • Figure 1: The fundamental quandle relations at a crossing and at a vertex.
  • Figure 2: Multiplying paths $\alpha$ and $\beta$ in $T(L)$ to form $\alpha \rhd \beta$.
  • Figure 3: $\lambda(\gamma) = a^{b\bar{d}}.$
  • Figure 4: Homotopy pushing $p$ to a neighboring arc of the edge.
  • Figure 5: Homotopy pushing $\gamma$ off an arc of $G$.
  • ...and 9 more figures

Theorems & Definitions (20)

  • Lemma 2.1
  • Definition
  • Theorem 2.2
  • proof
  • Definition
  • Corollary 3.1
  • Theorem 3.2
  • proof
  • Definition
  • Theorem 3.3
  • ...and 10 more