Title : Acylindrical surfaces in knot complements
(a joint work with Max Neumann-Coto)
Speaker : Mario Eudave-Mu\~noz (Universidad Nacional Autonoma de Mexico)
Abstract :

We consider acylindrical surfaces in closed 3-manifolds and in the complement of knots and links in the 3-sphere, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in a triangulation of the manifold and by the number of rational (or alternating) tangles in a projection of a link (or knot). These are topological interpretations of a theorem of Hass, which says that there is a bound for the genus of an acylindrical surface in a hyperbolic 3-manifold in terms only of volume. We construct for each $g$ examples of knots and links with tunnel number 2 and manifolds of Heegaard genus 3 which contain acylindrical surfaces of genus $g$, showing that there is no bound for the genus of an acylindrical surface in terms of tunnel number or Heegaard genus. We also give examples of 3-bridge knots containing quasi-Fuchsian surfaces of arbitrarily high genus, so the genus of this kind of surfaces is not bounded in terms of volume, crossing number or the number of tetrahedra.

pdf-file