#### Volume 10, issue 2 (2010)

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Bridge number and Conway products

### Ryan C Blair

Algebraic & Geometric Topology 10 (2010) 789–823
##### 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.

##### Keywords
bridge position, knot, Conway product
##### Mathematical Subject Classification 2000
Primary: 57M25, 57M27, 57M50