A plumbing-multiplicative function from the Links-Gould invariant

TL;DR

This work extends the Alexander-polynomial intuition to the non-semisimple Links-Gould invariant by proving that the top coefficient of $LG(L;p,q)$, when evaluated on the boundary of a Seifert surface, is multiplicative under plumbing. The authors develop and leverage a genus bound and a cubic skein relation to show that the top-term behaves like a product under annular plumbing, which in turn yields strong topological consequences: the LG invariant of fibred links is $\mathbb{Z}[q^{\pm1}]$-monic, and annular surfaces bound by links impose rigid multiplicative constraints on the top coefficient. These results lead to a plumbing-uniqueness statement and obstructions for links to bound annular surfaces, together with concrete computations that test the theory on small knots and highlight cases where annular surfaces do not exist. The paper thus connects non-semisimple quantum invariants with explicit topological questions, offering a q-deformed analogue of classical Alexander-type properties and inviting further exploration in Murasugi-type sums and related invariants.

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.

Figures (19)

• Figure 1: Plumbing surfaces $\Sigma$ and $\Sigma'$ along arcs $\sigma,\sigma'$.
• Figure 2: The unknotted twisted annulus $A_{-2}$.
• Figure 3: A Hopf link of annuli.
• Figure 4: Left: the surface $\Sigma$ obtained from plumbing $A_1$ onto $\Sigma_2$. The other surfaces arise from applying the cubic skein relation.
• Figure 5: The surface $\Sigma$ is obtained from gluing together the two arcs on the boundary of $\Sigma_0$. The tangle $T_\Sigma$ is also shown.

Theorems & Definitions (26)

• Theorem 1
• Corollary 1
• Corollary 2
• Corollary 3: Plumbing-uniqueness
• Lemma 3.1
• proof
• Lemma 3.2: Ishii:algebraic
• Theorem 3.3
• proof : Proof Theorem \ref{['theorem: genus bound for LG']}
• Example 3.4