Bridge number and Conway products

Abstract

In this paper, we give a structure theorem for c-incompressible Conway spheres in link complements in terms of the standard height function on $S^3$. We go on to define the generalized Conway product $K_1{\ast }_c{K}_2$ of two links $K_1$ and $K_2$. Provided $K_1{\ast }_c{K}_2$ satisfies minor additional hypotheses, we prove the lower bound $\beta \left(K_1{\ast }_c{K}_2\right)\ge \beta \left(K_1\right)-1$ for the bridge number of the generalized Conway product where $K_1$ is the distinguished factor. Finally, we present examples illustrating that this lower bound is tight.