Band taut sutured manifolds

TL;DR

This paper introduces band taut sutured manifolds and demonstrates their utility in understanding tunnel structures in knots, with applications to the cabling conjecture and properties of band sums.

Contribution

It establishes the concept of band taut sutured manifolds and proves their hierarchies, linking essential surfaces, sutures, and 2-handle attachments in 3-manifolds.

Findings

Tunnels for tunnel number one knots can be isotoped on branched surfaces.

Band sums satisfy the cabling conjecture.

Unknotting number one knots are prime and genus is superadditive under band sum.

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.