Torres-type formulas for link signatures

TL;DR

This work studies how the multivariable link signature $σ_L$ behaves as a coordinate approaches 1, presenting a 3D approach based on C-complexes and a 4D approach via a Torres-type extension to the full torus. It proves a Torres-type extension of the signature and nullity to $𝕋^μ$, together with explicit formulas relating the full-torus values to those of sub-links, and introduces slopes and Maslov-index corrections that control boundary gluing phenomena. A central result is a 3D limit theorem giving sharp bounds and, in many cases, exact limits for $σ_L(ω)$ as one coordinate tends to 1, with a complementary 4D analysis yielding broader limit statements for all coordinates tending to 1. The paper also connects these limits to the Levine-Tristram signature, provides new inequalities involving the Alexander module, and clarifies how linking data governs the limiting behavior, with implications for estimating the four-genus and related invariants. Overall, the work furnishes natural Torres-type extensions, precise 3D and 4D limit formulas, and a coherent framework linking multivariable and one-variable signature theories.

Abstract

We investigate the limits of the multivariable signature function $σ_L$ of a $μ$-component link $L$ as some variable tends to $1$ via two different approaches: a three-dimensional and a four-dimensional one. The first uses the definition of $σ_L$ by generalized Seifert surfaces and forms. The second relies on a new extension of $σ_L$ from its usual domain $(S^1\setminus\{1\})^μ$ to the full torus $\mathbb{T}^μ$ together with a Torres-type formula for $σ_L$, results which are of independent interest. Among several consequences, we obtain new estimates on the value of the Levine-Tristram signature of a link close to $1$.

Figures (9)

• Figure 1: A clasp intersection.
• Figure 2: The link $L(k)$, together with an associated C-complex, in the case $k=2$.
• Figure 3: The link $T(2,2\ell)$ (here with $\ell=3$) together with an associated C-complex.
• Figure 4: The values of $\sigma_L$ for $L=T(2,2\ell)$ with $\ell=3$, on the open torus $\mathbb{T}^2_*\simeq(0,1)^2$. The function $\eta_L$ is equal to $1$ on the diagonals, and vanishes everywhere else.
• Figure 5: The setting of the Novikov-Wall theorem.

Theorems & Definitions (121)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Example 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Lemma 2.1