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Multivariate Alexander quandles, V. Constructing the medial quandle of a link

Lorenzo Traldi

TL;DR

The paper shows that the medial quandle MQ(L) of a link L is determined by the enhanced reduced Alexander module M_A^{enr}(L), and, in the knot case, MQ(L) recovers the standard Alexander quandle on ker φ_L^{red}. It provides a concrete construction Q(N,(m_i),(X_i)) from a Λ-module framework and proves that every medial quandle is isomorphic to one of these, linking quandle theory to the displacements of the Alexander module. It further analyzes the relationships between MQ(L), the maximal semiregular image Q_A^{red}(L), and the involutory medial quandle IMQ(L), showing MQ(L) is generally weaker than M_A^{enr}(L) for multi-component links. The paper also offers detailed examples (Whitehead's links, 7^2_8, Hopf links) to illustrate how these invariants can coincide at some levels while diverging at others, highlighting the relative strengths of enhanced Alexander data, MQ, and IMQ in distinguishing sublinks and link types.

Abstract

We explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module.

Multivariate Alexander quandles, V. Constructing the medial quandle of a link

TL;DR

The paper shows that the medial quandle MQ(L) of a link L is determined by the enhanced reduced Alexander module M_A^{enr}(L), and, in the knot case, MQ(L) recovers the standard Alexander quandle on ker φ_L^{red}. It provides a concrete construction Q(N,(m_i),(X_i)) from a Λ-module framework and proves that every medial quandle is isomorphic to one of these, linking quandle theory to the displacements of the Alexander module. It further analyzes the relationships between MQ(L), the maximal semiregular image Q_A^{red}(L), and the involutory medial quandle IMQ(L), showing MQ(L) is generally weaker than M_A^{enr}(L) for multi-component links. The paper also offers detailed examples (Whitehead's links, 7^2_8, Hopf links) to illustrate how these invariants can coincide at some levels while diverging at others, highlighting the relative strengths of enhanced Alexander data, MQ, and IMQ in distinguishing sublinks and link types.

Abstract

We explain how the medial quandle of a classical or virtual link can be built from the peripheral structure of the reduced Alexander module.
Paper Structure (11 sections, 31 theorems, 106 equations, 6 figures)

This paper contains 11 sections, 31 theorems, 106 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Medial Quandles
  3. Definition \ref{['qquandle']} and Theorem \ref{['main1']}
  4. Notation and Terminology Regarding Links
  5. Proof of Theorem \ref{['main']}
  6. Semiregular Medial Quandles
  7. Involutory Medial Quandles
  8. Examples
  9. Whitehead's link
  10. The link $7^2_8$
  11. The classical and virtual Hopf links

Key Result

Theorem 2

If $Q$ is a medial quandle, then there are $M,N,I, (m_i)$ and $(X_i)$ such that $Q \cong Q(N,(m_i),(X_i))$.

Figures (6)

  • Figure 1: The underpassing arcs $b_1(c)$ and $b_2(c)$ of a classical crossing $c$ are on the right and left sides of the overpassing arc $a(c)$, respectively.
  • Figure 2: The writhe of a classical crossing $c$.
  • Figure 3: Indexing the arcs and classical crossings of a diagram with alternating writhes.
  • Figure 4: Diagrams $D$ and $D'$ of two versions of Whitehead's link, $W$ and $W'$.
  • Figure 5: A diagram $E$ of the link $L=7^2_8$.
  • ...and 1 more figures

Theorems & Definitions (59)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Definition 5
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • Lemma 8
  • ...and 49 more