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Peripheral elements in reduced Alexander modules: an addendum

Daniel S. Silver, Lorenzo Traldi

TL;DR

This addendum answers whether the sum of the longitudes $\sum_i\chi_i(L)$ vanishes in the reduced Alexander module for classical links, affirming it as $0$ via a Seifert-surface argument and standard covering-space reasoning, while highlighting that virtual links can exhibit nonzero sums. It further clarifies how the vanishing result extends under certain covering constructions and checkerboard colorability, and provides a corrected algebraic statement to ensure the peripheral structure governs linking numbers, aligning with Sakuma's classical result. The work sharpens the understanding of the peripheral structure in the reduced Alexander module and clarifies the classical–virtual dichotomy in this invariant.

Abstract

We answer a question raised in ``Peripheral elements in reduced Alexander modules'' [J. Knot Theory Ramifications 31 (2022), 2250058]. We also correct a minor error in that paper.

Peripheral elements in reduced Alexander modules: an addendum

TL;DR

This addendum answers whether the sum of the longitudes vanishes in the reduced Alexander module for classical links, affirming it as via a Seifert-surface argument and standard covering-space reasoning, while highlighting that virtual links can exhibit nonzero sums. It further clarifies how the vanishing result extends under certain covering constructions and checkerboard colorability, and provides a corrected algebraic statement to ensure the peripheral structure governs linking numbers, aligning with Sakuma's classical result. The work sharpens the understanding of the peripheral structure in the reduced Alexander module and clarifies the classical–virtual dichotomy in this invariant.

Abstract

We answer a question raised in ``Peripheral elements in reduced Alexander modules'' [J. Knot Theory Ramifications 31 (2022), 2250058]. We also correct a minor error in that paper.
Paper Structure (3 sections, 1 theorem, 2 equations)

This paper contains 3 sections, 1 theorem, 2 equations.

Table of Contents

  1. Introduction
  2. Virtual links
  3. Two additional comments on T

Key Result

Lemma 3.1

Let $L=K_1 \cup \dots \cup K_{\mu}$ be a classical or virtual link. If $i \in \{1, \dots, \mu\}$, then $\epsilon^\mu \varphi_L (\chi_i(L))$ is the element of $\mathbb Z ^ \mu$ whose $j$th coordinate is $0$ if $j=1$, $\ell_{j/i}(K_i,K_j)$ if $i \neq j \neq 1$, and if $i=j \neq 1$.

Theorems & Definitions (1)

  • Lemma 3.1