Attaching a –handle
to a genus two or greater boundary component of a
–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
–manifold, the number
of intersections between the boundary of the surface and one of the sutures, and the cocore of the
–handle in the manifold
after attaching a –handle
along the suture. We use this result to show that tunnels for tunnel number one knots or links in
any –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.
