Closed quasi-Fuchsian surfaces in hyperbolic knot complements

Hyperbolic Knot Complements without Closed Embedded Totally Geodesic Surfaces

It is conjectured that a hyperbolic knot complement does not contain a closed embedded totally geodesic surface. In this paper, we show that there are no such surfaces in the complements of hyperbolic 3-bridge knots and double torus knots. Some topological criteria for a closed essential surface failing to be totally geodesic are given. Roughly speaking, sufficiently ‘complicated’ surfaces can ...

Geodesic Surfaces in Knot Complements

Complex hyperbolic quasi-Fuchsian groups

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current st...

Noncompact Fuchsian and quasi-Fuchsian surfacesin hyperbolic 3--manifolds

Given a noncompact quasi-Fuchsian surface in a finite volume hyperbolic 3–manifold, we introduce a new invariant called the cusp thickness, that measures how far the surface is from being totally geodesic. We relate this new invariant to the width of a surface, which allows us to extend and generalize results known for totally geodesic surfaces. We also show that checkerboard surfaces provide e...

Generic Hyperbolic Knot Complements without Hidden Symmetries

We describe conditions under which, for a two-component hyperbolic link in S3 with an unknotted component, one can show that all but finitely many hyperbolic knots obtained by 1/n-surgery on that component lack hidden symmetries. We apply these conditions to the two-component links through nine crossings in the Rolfsen table, and to various other links in the literature. In each case we show th...