Bridge Number and Conway Products

TL;DR

This paper introduces a generalized Conway product for links, establishing a lower bound on the bridge number and demonstrating its tightness for an infinite class of links with high bridge numbers.

Contribution

It defines the generalized Conway product and proves a new lower bound on the bridge number, extending previous results.

Findings

Lower bound on bridge number: (K_{1} _{c} K_{2})  (K_{1}) - 1

The lower bound is tight for infinitely many links with high bridge numbers

Extends previous work on Conway products and bridge numbers

Abstract

Schubert proved that, given a composite link $K$ with summands $K_{1}$ and $K_{2}$, the bridge number of $K$ satisfies the following equation: $$\beta(K)=\beta(K_{1})+\beta(K_{2})-1.$$ In ``Conway Produts and Links with Multiple Bridge Surfaces", Scharlemann and Tomova proved that, given links $K_{1}$ and $K_{2}$, there is a Conway product $K_{1}\times_{c}K_{2}$ such that $$\beta(K_{1}\times_{c} K_{2}) \leq \beta(K_{1}) + \beta(K_{2}) - 1$$ In this paper, we define the generalized Conway product $K_{1}\ast_{c}K_{2}$ and prove the lower bound $\beta(K_{1}\ast_{c}K_{2}) \geq \beta(K_{1})-1$ where $K_{1}$ is the distinguished factor of the generalized product. We go on to show this lower bound is tight for an infinite class of links with arbitrarily high bridge number.