Geometry & Topology

The sutured Floer homology polytope

András Juhász

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In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality and then define a polytope P(M,γ) in H2(M,M;) that is spanned by the Spinc–structures which support nonzero Floer homology groups. If (M,γ)(M,γ) is a taut surface decomposition, then an affine map projects P(M,γ) onto a face of P(M,γ); moreover, if H2(M)=0, then every face of P(M,γ) can be obtained in this way for some surface decomposition. We show that if (M,γ) is reduced, horizontally prime and H2(M)=0, then P(M,γ) is maximal dimensional in H2(M,M;). This implies that if rk(SFH(M,γ))<2k+1, then (M,γ) has depth at most 2k. Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots.

Article information

Geom. Topol., Volume 14, Number 3 (2010), 1303-1354.

Received: 16 February 2010
Revised: 2 April 2010
Accepted: 3 May 2010
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

sutured manifold Heegaard Floer homology knot theory


Juhász, András. The sutured Floer homology polytope. Geom. Topol. 14 (2010), no. 3, 1303--1354. doi:10.2140/gt.2010.14.1303.

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