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The sutured Floer homology polytope

András Juhász

Geometry & Topology 14 (2010) 1303–1354

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.

sutured manifold, Heegaard Floer homology, knot theory
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 57R58
Received: 16 February 2010
Revised: 2 April 2010
Accepted: 3 May 2010
Published: 31 May 2010
Proposed: David Gabai
Seconded: Peter Ozsváth, Tom Mrowka
András Juhász
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB