The Levine-Tristram signature: a survey

TL;DR

This survey reviews the Levine-Tristram signature, an invariant for oriented links in 3-spheres, detailing its definitions, properties, and applications in link theory and concordance.

Contribution

It compiles and explains the various definitions, properties, and applications of the Levine-Tristram signature, providing a comprehensive reference for researchers.

Findings

Summarizes three and four dimensional definitions of the signature.

Lists key properties and applications in link theory.

Provides extensive references for proofs and further study.

Abstract

The Levine-Tristram signature associates to each oriented link $L$ in $S^3$ a function $\sigma_L \colon S^1 \to \mathbb{Z}.$ This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking numbers and link concordance. In this survey, we recall the three and four dimensional definitions of $\sigma_L$, list its main properties and applications, and give comprehensive references for the proofs of these statements.