A plumbing-multiplicative function from the Links-Gould invariant

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.