Bridge spectra of twisted torus knots

TL;DR

This paper investigates the bridge spectra of hyperbolic twisted torus knots, revealing large gaps and providing new examples of knots with complex bridge properties, and also addresses related problems in tunnel number one knots.

Contribution

It introduces the first examples of hyperbolic knots with arbitrarily large gaps in their bridge spectra and solves open problems related to tunnel number one knots.

Findings

Bridge spectra of certain hyperbolic twisted torus knots have arbitrarily large gaps.

Existence of Berge and Dean knots with arbitrarily large genus one bridge numbers.

Provides solutions to problems of Eudave-Muñoz concerning tunnel number one knots.

Abstract

We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbolic knots exhibiting this property. In addition, we show that there are Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a result, we give solutions to problems of Eudave-Mu\~noz concerning tunnel number one knots.