Acylindrical surfaces in 3-manifolds and knot complements

Details Acylindrical surfaces in 3-manifolds and knot complements Acylindrical surfaces in 3-manifolds and knot complements

2006-03-24

arxiv.org/abs/math/0603589v1

Boletin de la Sociedad Matematica Mexicana (3) vol. 10, special isue (2004), 147-169

Mathematics

Geometric Topology

22 pages, 14 figures. Boletin de la Sociedad Matematica Mexicana, special issue in honor of Francisco Gonzalez Acuna

Scientific paper

We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of rational (or alternating) tangles in a projection of a link (or knot). For each g we find knots with tunnel number 2 and manifolds of Heegaard genus 3 containing acylindrical surfaces of genus g. Finally, we construct 3-bridge knots containing quasi-Fuchsinan surfaces of unbounded genus, and use them to find manifolds of Heegaard genus 2 and homology spheres of Heegaard genus 3 containing infinitely many incompressible surfaces.

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