Volume 14, issue 1 (2014)

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ISSN (electronic): 1472-2739
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Band-taut sutured manifolds

Scott A Taylor

Algebraic & Geometric Topology 14 (2014) 157–215

Attaching a 2–handle to a genus two or greater boundary component of a 3–manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3–manifold, the number of intersections between the boundary of the surface and one of the sutures, and the cocore of the 2–handle in the manifold after attaching a 2–handle along the suture. We use this result to show that tunnels for tunnel number one knots or links in any 3–manifold can be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the knot or link exterior. In a subsequent paper, we use the theorem to prove that band sums satisfy the cabling conjecture, and to give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. To prove the theorem, we introduce band-taut sutured manifolds and prove the existence of band-taut sutured manifold hierarchies.

$3$–manifold, sutured manifold, $2$–handle addition, tunnel
Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 57M25
Received: 23 August 2012
Revised: 12 June 2013
Accepted: 15 July 2013
Preview posted: 21 November 2013
Published: 9 January 2014
Scott A Taylor
Department of Mathematics and Statistics
Colby College
5832 Mayflower Hill
Waterville, ME 04901