Extending fibrations of knot complements to ribbon disk 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...

Geodesic Surfaces in Knot Complements

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 ...

Distance of Heegaard Splittings of Knot Complements

Let K be a knot in a closed orientable irreducible 3-manifold M and let P be a Heegaard splitting of the knot complement of genus at least two. Suppose Q is a bridge surface for K. Then either • d(P ) ≤ 2 − χ(Q −K), or • K can be isotoped to be disjoint from Q so that after the isotopy Q is a Heegaard surface for the knot exterior that is isotopic to a possibly stabilized copy of P .