## Algebraic & Geometric Topology

### On sutured Floer homology and the equivalence of Seifert surfaces

#### 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 $S3(R)$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 1 (2013), 505-548.

Dates
Accepted: 7 October 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715505

Digital Object Identifier
doi:10.2140/agt.2013.13.505

Mathematical Reviews number (MathSciNet)
MR3116378

Zentralblatt MATH identifier
1272.57008

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57R58: Floer homology

#### Citation

Hedden, Matthew; Juhász, András; Sarkar, Sucharit. On sutured Floer homology and the equivalence of Seifert surfaces. Algebr. Geom. Topol. 13 (2013), no. 1, 505--548. doi:10.2140/agt.2013.13.505. https://projecteuclid.org/euclid.agt/1513715505

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