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On sutured Floer homology and the equivalence of Seifert surfaces

Matthew Hedden, András Juhász and Sucharit Sarkar

Algebraic & Geometric Topology 13 (2013) 505–548

The goal of this paper is twofold. First, given a Seifert surface R in the 3–sphere, we show how to construct a Heegaard diagram for the sutured manifold S3(R) complementary to R, which in turn enables us to compute the sutured Floer homology of S3(R) combinatorially. Secondly, we outline how the sutured Floer homology of S3(R), together with the Seifert form of R, can be used to decide whether two minimal genus Seifert surfaces of a given knot are isotopic in S3. We illustrate our techniques by showing that the knot 83 has two minimal genus Seifert surfaces up to isotopy. Furthermore, for any n 1 we exhibit a knot Kn that has at least n nonisotopic free minimal genus Seifert surfaces.

Heegaard diagram, Seifert surface, Floer homology
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57R58
Received: 9 March 2011
Accepted: 7 October 2012
Published: 10 March 2013
Matthew Hedden
Department of Mathematics
Michigan State University
East Lansing, MI 48824
András Juhász
Department of Mathematics
South Kensington Campus, Imperial College
London SW7 2AZ
Sucharit Sarkar
Department of Mathematics
Princeton University
Princeton, NJ 08544