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.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Burton, Stephan Daniel, "Unknotting Tunnels of Hyperbolic Tunnel Number n Manifolds" (2012). Theses and Dissertations. 3307.
Hyperbolic Geometry, Hyperbolic 3-manifolds, Unknotting Tunnel, Ford Domain, Knot Theory