Volume 13, issue 5 (2013)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Obtaining genus $2$ Heegaard splittings from Dehn surgery

Kenneth L Baker, Cameron Gordon and John Luecke

Algebraic & Geometric Topology 13 (2013) 2471–2634
Abstract

Let K be a hyperbolic knot in S3 and suppose that some Dehn surgery on K with distance at least 3 from the meridian yields a 3–manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck’s surface (the closed nonorientable surface of Euler characteristic 1), then the knot dual to the surgery is either 0–bridge or 1–bridge with respect to a genus 2 Heegaard splitting of M. In the case that M does contain an embedded Dyck’s surface, we obtain similar results. As a corollary, if M does not contain an incompressible genus 2 surface, then the tunnel number of K is at most 2.

Keywords
Dehn surgery, bridge number, Heegaard splitting
Mathematical Subject Classification 2010
Primary: 57M27
References
Publication
Received: 1 May 2012
Revised: 6 February 2013
Accepted: 7 March 2013
Published: 5 July 2013
Authors
Kenneth L Baker
Department of Mathematics
University of Miami
PO Box 249085
Coral Gables, FL 33146
USA
http://math.miami.edu/~kenken
Cameron Gordon
Department of Mathematics
University of Texas at Austin
1 University Station C1200
Austin, TX 78712-1202
USA
http://www.ma.utexas.edu/text/webpages/gordon.html
John Luecke
Department of Mathematics
The University of Texas at Austin
1 University Station C1200
Austin, TX 78712-0257
USA