Extended signatures and link concordance

The Levine-Tristram signature admits an n-variable extension for n-component links: it was first defined as an integer valued function on $(S^1\setminus{1})^n$, and recently extended to the full torus $T^n$. The aim of the present article is to study and use this extended signature. First, we show that it is constant on the connected components of the complement of the zero-locus of some renormalized Alexander polynomial. Then, we prove that the extended signature is a concordance invariant on an explicit dense subset of $T^n$. Finally, as an application, we present an infinite family of 3-component links with the following property: these links are not concordant to their mirror image, a fact that can be detected neither by the non-extended signatures, nor by the multivariable Alexander polynomial, nor by the Milnor triple linking number.

💡 Research Summary

The paper studies an extension of the classical Levine‑Tristram signature to the full µ‑torus for coloured links, and uses this “extended signature” to obtain new concordance invariants.

Background. For an oriented µ‑coloured link L the Levine‑Tristram signature σL(ω) is originally defined only for ω∈(S¹∖{1})^µ. It is constant on the connected components of the complement of the zero set of the multivariable Alexander polynomial ΔL, and it is known to be invariant under topological concordance on a dense subset of the torus where ω does not annihilate any polynomial p∈ℤ