Mark Brittenham

PERSISTENTLY LAMINAR TANGLES

We show how to build tangles T in a 3-ball with the property that any knot obtained by tangle sum with T has a persistent lamination in its exterior, and therefore has property P. The construction is based on an example of a persistent lamination in the exterior of the twist knot 61, due to Ulrich Oertel. We also show how the construction can be generalized to n-string tangles.

Exceptional Seifert-fibered spaces and Dehn surgery on 2-bridge knots

Abstract We show that non-integer surgery on a non-torus 2-bridge knot can never yield an exceptional Scifert-fibered space. In most cases, no surgery will yield an exceptional Scifert-fibered space.

A uniform model for almost convexity and rewriting systems

Abstract We introduce a topological property for finitely generated groups, called stackable, that implies the existence of an inductive procedure for constructing van Kampen diagrams with respect to a particular finite presentation. We also define algorithmically stackable groups, for which this procedure is an algorithm. This property gives a common model for algorithms arising from both rewriting systems and almost convexity for groups.

Persistent laminations from Seifert surfaces

We show how an incompressible Seifert surface F for a knot K in S3 can be used to create an essential lamination ℒF in the complement of each of an infinite class of knots associated to F. This lamination is persistent for these knots; it remains essential under all non-trivial Dehn fillings of the knot complement. This implies a very strong form of Property P for each of these knots.

Essential laminations and deformations of homotopy equivalences: from essential pullback to homeomorphism

Abstract In this paper we develop techniques for using essential laminations to deform homotopy equivalences of 3-manifolds to homeomorphisms.

Essential laminations in non-Haken 3-manifolds

Abstract In this paper we show that an essential lamination L in a non-Haken 3-manifold M is “tightly wrapped” in the sense that any two leaves of L have intersecting closures; L therefore contains a unique minimal sublamination. We also show that these properties are inherited by any lift of L to a finite cover of M.

Tame filling invariants for groups

A new pair of asymptotic invariants for finitely presented groups, called intrinsic and extrinsic tame filling functions, is introduced. These filling functions are quasi-isometry invariants that strengthen the notions of intrinsic and extrinsic diameter functions for finitely presented groups. We show that the existence of a (finite-valued) tame filling functions implies that the group is tame combable. Bounds on both intrinsic and extrinsic tame filling functions are discussed for stackable groups, including groups with a finite complete rewriting system, Thompsons group F, and almost convex groups.

Essential laminations in Seifert-fibered spaces: Boundary behavior

Abstract We show that an essential lamination in a Seifert-fibered space M rarely meets the boundary of M in a Reeb-foliated annulus.

Geometry of the word problem for 3-manifold groups

Abstract We provide an algorithm to solve the word problem in all fundamental groups of 3-manifolds that are either closed, or compact with (finitely many) boundary components consisting of incompressible tori, by showing that these groups are autostackable. In particular, this gives a common framework to solve the word problem in these 3-manifold groups using finite state automata. We also introduce the notion of a group which is autostackable respecting a subgroup, and show that a fundamental group of a graph of groups whose vertex groups are autostackable respecting any edge group is autostackable. A group that is strongly coset automatic over an autostackable subgroup, using a prefix-closed transversal, is also shown to be autostackable respecting that subgroup. Building on work by Antolin and Ciobanu, we show that a finitely generated group that is hyperbolic relative to a collection of abelian subgroups is also strongly coset automatic relative to each subgroup in the collection. Finally, we show that fundamental groups of compact geometric 3-manifolds, with boundary consisting of (finitely many) incompressible torus components, are autostackable respecting any choice of peripheral subgroup.

KNOTS WITH UNIQUE MINIMAL GENUS SEIFERT SURFACE AND DEPTH OF KNOTS

We describe a procedure for creating infinite families of hyperbolic knots, each having unique minimal genus Seifert surface which cannot be the sole compact leaf of a depth one foliation.