We demonstrate that for all but a finite number of Dehn fillings on a cusped manifold, the core of the attached solid torus is isotopic into every Heegaard surface for the filled manifold. Furthermore, if the cusped manifold does not contain a closed, non-peripheral, incompressible surface, then after excluding the aforementioned set and those filled manifolds containing incompressible surfaces (also a finite set) every other manifold obtained by Dehn filling contains at most a finite number of Heegaard surfaces that are not Heegaard surfaces for the cusped manifold. It follows that these manifolds contain a finite number of Heegaard surfaces of bounded genera. For each cusped manifold, the excluded manifolds are contained in a finite set that can be determined algorithmically.

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.

Abstract. Given a knot K in a closed orientable manifold M we define the growth rate of the tunnel number of K to be gr t (K) = lim sup n→∞ t(nK)−nt(K) n−1. As our main result we prove that the Heegaard genus of M is strictly less than the Heegaard genus of the knot exterior if and only if the growth rate is less than 1. In particular this shows that a non-trivial knot in S 3 is never asymptotically super additive. The main result gives conditions that imply falsehood of Morimoto's Conjecture.

We prove that deciding if a diagram of the unknot can be untangled using at most k Reidemeister moves (where k is part of the input) is NP-hard. We also prove that several natural questions regarding links in the 3-sphere are NP-hard, including detecting whether a link contains a trivial sublink with n components, computing the unlinking number of a link, and computing a variety of link invariants related to four-dimensional topology (such as the 4-ball Euler characteristic, the slicing number, and the 4-dimensional clasp number).

Abstract. We prove that if K 1 ⊂ M 1 , . . . , K n ⊂ M n are m-small knots in closed orientable 3-manifolds then the Heegaard genus of E(# n i=1 K i ) is strictly less than the sum of the Heegaard genera of the E(K i ) (i = 1, . . . , n) if and only if there exists a proper subset I of {1, . . . , n} so that # i∈I K i admits a primitive meridian. This generalizes the main result of Morimoto in [17].