Knot complements and groups

Cofinitely Hopfian Groups, Open Mappings and Knot Complements

A group Γ is defined to be cofinitely Hopfian if every homomorphism Γ → Γ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivi...

Floer Homology and Knot Complements

We use the Ozsváth-Szabó theory of Floer homology to define an invariant of knot complements in three-manifolds. This invariant takes the form of a filtered chain complex, which we call ĈF r. It carries information about the Ozsváth-Szabó Floer homology of large integral surgeries on the knot. Using the exact triangle, we derive information about other surgeries on knots, and about the maps on ...

Geodesic Surfaces in Knot Complements

Geometric Limits of Knot Complements

We prove that any complete hyperbolic 3–manifold with finitely generated fundamental group, with a single topological end, and which embeds into S is the geometric limit of a sequence of hyperbolic knot complements in S. In particular, we derive the existence of hyperbolic knot complements which contain balls of arbitrarily large radius. We also show that a complete hyperbolic 3–manifold with t...

Simplicial Structures of Knot Complements

It was shown in [5] that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3-manifold which contains no fibred sub-manifolds as strongly simple pieces of its JSJ-decomposition. In this paper we prove a generalisation of that result to all knot complements. The explicit formula for the bound is in terms of the numbers of tetrahedr...