Knots: Topology and H 3 d H 3 Essay example

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Geometry & Topology 12 (2008) 2095–2171 2095
Closed quasi-Fuchsian surfaces in hyperbolic knot complements
JOSEPH D MASTERS XINGRU ZHANG
We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.
57N35; 57M25
1 Introduction
By a knot complement we mean, in this paper, the complement of a knot in a connected closed orientable 3–manifold (which is not necessarily S3). A knot complement is said to be hyperbolic if it admits a complete hyperbolic metric of finite volume. By a surface we mean, in this paper, the complement of a finite (possibly empty) set of points in the interior of a compact, connected, orientable 2–manifold. By a surface in a 3–manifold M; we mean a continuous, proper map f W S ! M from a surface S into M.AsurfacefWS!M ina3–manifoldM issaidtobeincompressibleifS isnot a 2–sphere and the induced homomorphism f W 1.S; s/ ! 1.M; f .s// is injective forone(andthusforany)choiceofbasepointsinS.AsurfacefWS!M ina 3–manifold M is said to be essential if it is incompressible and the map f W S ! M cannot be homotoped into a boundary component or an end component of M .
Essential surfaces in hyperbolic knot complements can be divided into three mutually exclusive geometric types: quasi-Fuchsian surfaces, geometrically infinite surfaces and essential surfaces with accidental parabolics. Now we recall the relevant terminology. Let H3 denote the hyperbolic 3–space (always in the upper half space model) and let S12 D C [ f1g denote the…