On the signatures of torus knots

TL;DR

This paper investigates the signature function of torus knots, providing elementary proofs and explicit formulas, including a closed form involving Dedekind sums, enhancing understanding of their mathematical properties.

Contribution

It offers a new elementary proof for the integral of the signature function and derives a closed formula for the Tristram-Levine signature of torus knots using Dedekind sums.

Findings

Elementary proof of the integral of the signature function

Closed formula for the Tristram-Levine signature involving Dedekind sums

Enhanced understanding of torus knot signatures

Abstract

We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.