The sutured Floer homology polytope

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,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decomposition, then a natural map projects P(M',g') onto a face of P(M,g); moreover, if H_2(M) = 0, then every face of P(M,g) can be obtained in this way for some surface decomposition. We show that if (M,g) is reduced, horizontally prime, and H_2(M) = 0, then P(M,g) is maximal dimensional in H^2(M,\partial M; R). This implies that if rk(SFH(M,g)) < 2^{k+1} then (M,g) 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.

37 pages, 4 figures, improved exposition

References in corpus (2)

• Knot Floer homology detects fibred knots
• Corrigendum to "Knot Floer homology detects fibred knots"

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• Heegaard Floer homology of spatial graphs
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• The sutured Floer polytope and taut depth one foliations
• The embedded contact homology of sutured solid tori
• Kauffman states and Heegaard diagrams for tangles
• Kuperberg invariants for balanced sutured 3-manifolds
• Symmetries and adjunction inequalities for knot Floer homology
• Spectral order for contact manifolds with convex boundary
• Floer homology and non-fibered knot detection
• Closures of $3$-braids and detection
• Nearly fibered links with genus one