Signature spectrum of positive braids

TL;DR

This paper establishes bounds on Levine-Tristram signatures of positive braid links, linking them to the first Betti number, and provides uniform bounds on signature ratios within positive braid monoids.

Contribution

It introduces a linear lower bound for all Levine-Tristram signatures of positive braid links based on the first Betti number, and derives uniform bounds on signature ratios.

Findings

Lower bound for signatures linear in Betti number

Uniform bounds on signature ratios in positive braid monoids

Enhanced understanding of signature invariants in braid theory

Abstract

We derive a lower bound for all Levine-Tristram signatures of positive braid links, linear in terms of the first Betti number. As a consequence, we obtain upper and lower bounds on the ratio of fixed pairs of Levine-Tristram signature invariants, valid uniformly on all positive braid monoids.