#### Volume 13, issue 1 (2013)

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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
##### Abstract

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 ${S}^{3}\left(R\right)$ complementary to $R$, which in turn enables us to compute the sutured Floer homology of ${S}^{3}\left(R\right)$ combinatorially. Secondly, we outline how the sutured Floer homology of ${S}^{3}\left(R\right)$, 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 ${S}^{3}$. We illustrate our techniques by showing that the knot ${8}_{3}$ has two minimal genus Seifert surfaces up to isotopy. Furthermore, for any $n\ge 1$ we exhibit a knot ${K}_{n}$ that has at least $n$ nonisotopic free minimal genus Seifert surfaces.

##### Keywords
Heegaard diagram, Seifert surface, Floer homology
Primary: 57M27
Secondary: 57R58