Limits of the Tristram--Levine signature function

TL;DR

This paper establishes a precise condition on the Alexander polynomial that determines the limit of the Tristram--Levine signature at 1 for links, linking it to the linking matrix.

Contribution

It provides a new criterion connecting the Alexander polynomial and the signature limit, clarifying the boundary of the signature function's behavior.

Findings

The limit at 1 of the signature is determined by the linking matrix under certain conditions.

A specific condition on the Alexander polynomial is identified.

The relationship between the Alexander polynomial and the signature limit is clarified.

Abstract

We show that under a precise condition on the single variable Alexander polynomial, the limit at $1$ of the Tristram--Levine signature of a link is determined by the linking matrix.