Obtaining genus 2 Heegaard splittings from Dehn surgery

Baker, Kenneth L¹ ; Gordon, Cameron² ; Luecke, John³

¹Department of Mathematics, University of Miami, PO Box 249085, Coral Gables, FL 33146, USA
²Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-1202, USA
³Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0257, USA

Algebraic and Geometric Topology, Tome 13 (2013) no. 5, pp. 2471-2634

Résumé

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.

Détail

DOI : 10.2140/agt.2013.13.2471

Classification : 57M27
Keywords: Dehn surgery, bridge number, Heegaard splitting

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