Bridge number and integral Dehn surgery

TL;DR

This paper investigates how twisting knots along an annulus affects their bridge number in 3-manifolds, establishing that the bridge number tends to infinity with the number of twists, especially for knots in Seifert fiber spaces with specific Dehn surgeries.

Contribution

The authors introduce a new framework for analyzing the growth of bridge numbers under twisting operations and provide bounds demonstrating their unbounded increase in certain 3-manifold contexts.

Findings

Bridge number of twisted knots tends to infinity as the number of twists increases.

For specific knots in Seifert fiber spaces, the bridge number grows without bound under Dehn surgeries.

The results contrast with existing conjectures about knots in lens spaces with 3-sphere surgeries.

Abstract

In a 3-manifold M, let K be a knot and R be an annulus which meets K transversely. We define the notion of the pair (R,K) being caught by a surface Q in the exterior of the link given by K and the boundary curves of R. For a caught pair (R,K), we consider the knot K^n gotten by twisting K n times along R and give a lower bound on the bridge number of K^n with respect to Heegaard splittings of M -- as a function of n, the genus of the splitting, and the catching surface Q. As a result, the bridge number of K^n tends to infinity with n. In application, we look at a family of knots K^n found by Teragaito that live in a small Seifert fiber space M and where each K^n admits a Dehn surgery giving the 3-sphere. We show that the bridge number of K^n with respect to any genus 2 Heegaard splitting of M tends to infinity with n. This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving the 3-sphere.