Bridge number and Conway products

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 $K_{1} *_{c} K_{2}$ of two links $K_{1}$ and $K_{2}$ . Provided $K_{1} *_{c} K_{2}$ satisfies minor additional hypotheses, we prove the lower bound $β (K_{1} *_{c} K_{2}) \geq β (K_{1}) - 1$ 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.

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Keywords

bridge position, knot, Conway product

Mathematical Subject Classification 2000

Primary: 57M25, 57M27, 57M50

References

Publication

Received: 13 May 2009

Revised: 15 December 2009

Accepted: 30 January 2010

Published: 31 March 2010

Authors

Ryan C Blair
Department of Mathematics University of California, Santa Barbara Santa Barbara, CA 93106

Volume 10, issue 2 (2010)

Ryan C Blair

Abstract