Diameter Bounds on the Complex of Minimal Genus Seifert Surfaces for Hyperbolic Knot

Author: Pelayo, Roberto Carlos

Year: 2007

Degree: Dissertation (Ph.D.)

Advisor: Calegari, Danny C.

Committee Members: Calegari, Danny C.; Ramakrishnan, Dinakar; Aschbacher, Michael; Dunfield, Nathan M.

Option: Mathematics

DOI: 10.7907/Q16J-V757

Abstract

Given a link L in the 3-sphere, one can build simplicial complexes MS(L) and IS(L), called the Kakimizu complexes. These complexes have isotopy classes of minimal genus and incompressible Seifert surfaces for L as their vertex sets and have simplicial structures defined via a disjointness property. The Kakimizu complexes enjoy many topological properties and are conjectured to be contractible. Following the work of Gabai on sutured manifolds and Murasugi sums, MS(L) and IS(L) have been classified for various classes of links. This thesis focuses on hyperbolic knots; using minimal surface representatives and Kakimizu's formulation of the path-metric on MS(K), we are able to bound the diameter of this complex in terms of only the genus of the knot. The techniques of this paper are also generalized to one-cusped manifolds with a preferred relative homology class.

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