Signature, positive Hopf plumbing and the Coxeter transformation

TL;DR

This paper investigates the spectral properties of Coxeter transformations associated with positive Hopf plumbings, establishing bounds on eigenvalues and signatures, and contrasting these with divide links.

Contribution

It provides new bounds on the signature of tree-like positive Hopf plumbings and explores the limitations of such bounds for divide links.

Findings

At least two thirds of Coxeter transformation eigenvalues lie on the unit circle.

The signature of tree-like positive Hopf plumbings can be bounded below by the genus.

For divide links, the signature cannot be linearly bounded from below by the genus.

Abstract

By a theorem of A'Campo, the eigenvalues of certain Coxeter transformations are positive real or lie on the unit circle. By optimally bounding the signature of tree-like positive Hopf plumbings from below by the genus, we prove that at least two thirds of them lie on the unit circle. In contrast, we show that for divide links, the signature cannot be linearly bounded from below by the genus.