Abstract

Adams conjectured that unknotting tunnels of tunnel number 1 manifolds are always isotopic to a geodesic. We generalize this question to tunnel number n manifolds. We find that there exist complete hyperbolic structures and a choice of spine of a compression body with genus 1 negative boundary and genus n ≥ 3 outer boundary for which (n−2) edges of the spine self-intersect. We use this to show that there exist finite volume one-cusped hyperbolic manifolds with a system of n tunnels for which (n−1) of the tunnels are homotopic to geodesics arbitrarily close to self-intersecting. This gives evidence that the generalization of Adam's conjecture to tunnel number n ≥ 2 manifolds may be false.

Degree

MS

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

http://lib.byu.edu/about/copyright/

Date Submitted

2012-07-02

Document Type

Thesis

Handle

http://hdl.lib.byu.edu/1877/etd5400

Keywords

Hyperbolic Geometry, Hyperbolic 3-manifolds, Unknotting Tunnel, Ford Domain, Knot Theory

Included in

Mathematics Commons

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