Bridge Number and Conway Products

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: $$β(K)=β(K_{1})+β(K_{2})-1.$$ 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 $$β(K_{1}\times_{c} K_{2}) \leq β(K_{1}) + β(K_{2}) - 1$$ In this paper, we define the generalized Conway product $K_{1}\ast_{c}K_{2}$ and prove the lower bound $β(K_{1}\ast_{c}K_{2}) \geq β(K_{1})-1$ 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.