Algebraic & Geometric Topology

Band-taut sutured manifolds

Scott A Taylor

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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.

Article information

Algebr. Geom. Topol., Volume 14, Number 1 (2014), 157-215.

Received: 23 August 2012
Revised: 12 June 2013
Accepted: 15 July 2013
First available in Project Euclid: 19 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

$3$–manifold sutured manifold $2$–handle addition tunnel


Taylor, Scott A. Band-taut sutured manifolds. Algebr. Geom. Topol. 14 (2014), no. 1, 157--215. doi:10.2140/agt.2014.14.157.

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