A spectrum connecting the braid index and the bridge index

TL;DR

This paper introduces the bridge-braid spectrum, a new invariant connecting braid and bridge indices, and provides formulas and data for prime knots up to 9 crossings.

Contribution

It defines the bridge-braid spectrum, linking classical invariants, and derives formulas for split and composite links, with comprehensive spectra tables for small prime knots.

Findings

The spectrum spans between the classical bridge and braid indices.

Formulas for spectra of split and composite links are established.

Spectra tables for all prime knots up to 9 crossings are generated.

Abstract

We study the booklink, a braid-like embedding with local maxima and minima, and the bridge-braid spectrum of a link, which captures the smallest number of braid-strands in a booklink with a prescribed number of critical points. This spectrum spans the gap between the classical bridge and braid indices. We apply a foliation theory argument to provide a formula for the spectra of both split and composite links. We then generate a table for the spectra of all prime knots up to 9 crossings.