Critical point theory in knot complements

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 ...

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...

C-incompressible Planar Surfaces in Knot Complements

In [6] Wu shows that if a link or a knot L in S3 in thin position has thin spheres, then the thin sphere of lowest width is an essential surface in the link complement. In this paper we show that if we further assume that L ⊂ S3 is prime, then the thin sphere of lowest width also does not have any cut-disks. We also prove an analogous result for a specific kind of tangles in