## Geometry & Topology

### The sutured Floer homology polytope

András Juhász

#### Abstract

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 $H2(M,∂M;ℝ)$ 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 $H2(M,∂M;ℝ)$. 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

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

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

https://projecteuclid.org/euclid.gt/1513732227

Digital Object Identifier
doi:10.2140/gt.2010.14.1303

Mathematical Reviews number (MathSciNet)
MR2653728

Zentralblatt MATH identifier
1236.57018

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

#### Citation

Juhász, András. The sutured Floer homology polytope. Geom. Topol. 14 (2010), no. 3, 1303--1354. doi:10.2140/gt.2010.14.1303. https://projecteuclid.org/euclid.gt/1513732227