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: $$\beta (K)=\beta (K_1)+\beta (K_2)-1.$$ 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 $$\beta (K_1{\times }_c{K}_2)\le \beta (K_1)+\beta (K_2)-1$$ In this paper, we define the generalized Conway product $K_1{\ast }_c{K}_2$ and prove the lower bound $\beta (K_1{\ast }_c{K}_2)\ge \beta (K_1)-1$ 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.

15 pages, 13 figures

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