## Algebraic & Geometric Topology

### Band-taut sutured manifolds

Scott A Taylor

#### Abstract

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

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

Dates
Revised: 12 June 2013
Accepted: 15 July 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715798

Digital Object Identifier
doi:10.2140/agt.2014.14.157

Mathematical Reviews number (MathSciNet)
MR3158757

Zentralblatt MATH identifier
1297.57041

#### Citation

Taylor, Scott A. Band-taut sutured manifolds. Algebr. Geom. Topol. 14 (2014), no. 1, 157--215. doi:10.2140/agt.2014.14.157. https://projecteuclid.org/euclid.agt/1513715798