A plumbing-multiplicative function from the Links-Gould invariant

TL;DR

This paper demonstrates that a specific Laurent polynomial derived from the Links-Gould invariant behaves multiplicatively under surface plumbing, leading to topological insights and obstructions related to fibred links and surface constructions.

Contribution

It establishes the multiplicativity of the top coefficient of the Links-Gould invariant under plumbing, and derives topological consequences for fibred links and surface bounding.

Findings

The Laurent polynomial's top coefficient is multiplicative under plumbing.

The Links-Gould invariant of fibred links is monic in Z[q^{\u00b1}].

Provides a topological obstruction for certain surface-bounding links.

Abstract

We prove that the Laurent polynomial in $\mathbb{Z}[q^{\pm 1}]$ that is the top coefficient of the Links-Gould invariant of the boundary of a Seifert surface is multiplicative under plumbing of surfaces. We deduce that the Links-Gould invariant of a fibred link in $S^3$ is $\mathbb{Z}[q^{\pm 1}]$-monic. As a purely topological application, we deduce a ``plumbing-uniqueness'' statement for links that bound surfaces obtained by plumbing/deplumbing unknotted twisted annuli as well as providing an obstruction for links to bound such surfaces.