Volume 13, issue 1 (2013)

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Dehn surgery on knots of wrapping number $2$

Ying-Qing Wu

Algebraic & Geometric Topology 13 (2013) 479–503

Suppose K is a hyperbolic knot in a solid torus V intersecting a meridian disk D twice. We will show that if K is not the Whitehead knot and the frontier of a regular neighborhood of K D is incompressible in the knot exterior, then K admits at most one exceptional surgery, which must be toroidal. Embedding V in S3 gives infinitely many knots Kn with a slope rn corresponding to a slope r of K in V . If r surgery on K in V is toroidal then either Kn(rn) are toroidal for all but at most three n, or they are all atoroidal and nonhyperbolic. These will be used to classify exceptional surgeries on wrapped Montesinos knots in a solid torus, obtained by connecting the top endpoints of a Montesinos tangle to the bottom endpoints by two arcs wrapping around the solid torus.

Exceptional Dhen Surgery, hyperbolic manifolds, wrapping number
Mathematical Subject Classification 2010
Primary: 57N10
Received: 25 September 2011
Revised: 24 July 2012
Accepted: 4 October 2012
Published: 6 March 2013
Ying-Qing Wu
Department of Mathematics
The University of Iowa
14 MacLean Hall
Iowa City, IA 52242-1419