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Peripheral elements in reduced Alexander modules

Lorenzo Traldi

TL;DR

The paper introduces peripheral elements in reduced Alexander modules for classical and virtual links, defining meridians $M_i(D)$ and longitudes $\chi_i(D)$ inside the reduced module $M_A^{\text{red}}(D)$ and packaging them into an enhanced module $M_A^{\text{enr}}(D)=(M_A^{\text{red}}(D),M_1(D),\dots,M_\mu(D),\chi_1(D),\dots,\chi_\mu(D))$. It proves that $M_A^{\text{enr}}$ is a link-type invariant under Reidemeister moves (and, notably, welded moves), and shows that this enhancement strictly increases sensitivity over $M_A^{\text{red}}$ by encoding all linking numbers via the map $\varphi_L$ and related constructions. The longitudes generate the $(1-t)$-torsion submodule, while for knots the enhancement is determined entirely by $\ker\varphi_K$, limiting its added value in that case. Through concrete examples such as Hopf and Borromean rings, the work demonstrates that $M_A^{\text{enr}}$ can distinguish links that share the reduced module data, establishing a stronger invariant with potential connections to Milnor invariants and medial quandle structures.

Abstract

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are $0$.

Peripheral elements in reduced Alexander modules

TL;DR

The paper introduces peripheral elements in reduced Alexander modules for classical and virtual links, defining meridians and longitudes inside the reduced module and packaging them into an enhanced module . It proves that is a link-type invariant under Reidemeister moves (and, notably, welded moves), and shows that this enhancement strictly increases sensitivity over by encoding all linking numbers via the map and related constructions. The longitudes generate the -torsion submodule, while for knots the enhancement is determined entirely by , limiting its added value in that case. Through concrete examples such as Hopf and Borromean rings, the work demonstrates that can distinguish links that share the reduced module data, establishing a stronger invariant with potential connections to Milnor invariants and medial quandle structures.

Abstract

We discuss meridians and longitudes in reduced Alexander modules of classical and virtual links. When these elements are suitably defined, each link component will have many meridians, but only one longitude. Enhancing the reduced Alexander module by singling out these peripheral elements provides a significantly stronger link invariant. In particular, the enhanced module determines all linking numbers in a link; in contrast, the module alone does not even detect how many linking numbers are .
Paper Structure (7 sections, 9 theorems, 68 equations, 10 figures)

This paper contains 7 sections, 9 theorems, 68 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Linking numbers
  4. Longitudes and torsion
  5. Knots
  6. Examples
  7. Some questions

Key Result

Theorem 5

Let $D$ and $D'$ be equivalent link diagrams. Then there is an isomorphism $f:M_A^{\textup{enr}}(D) \to M_A^{\textup{enr}}(D')$.

Figures (10)

  • Figure 1: A classical crossing $c$.
  • Figure 2: The writhe of a classical crossing $c$.
  • Figure 3: The welded move.
  • Figure 4: An $\Omega.1$ move changes $D$ into $D'$.
  • Figure 5: An $\Omega.1$ move changes $D$ into $D'$.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Definition 6
  • Theorem 7
  • Lemma 8
  • Theorem 9
  • proof
  • ...and 10 more