Hoomorphic disks and genus bounds

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243 Scopus citations


We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to new proofs of certain results previously obtained using Seiberg-Witten monopole Floer homology (in collaboration with Kronheimer and Mrowka). It also leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof shows that the canonical element of Heegaard Floer homology associated to a weakly symplectically fillable contact structure is non-trivial. In particular, for certain three-manifolds, Heegaard Floer homology gives obstructions to the existence of taut foliations.

Original languageEnglish (US)
Pages (from-to)311-334
Number of pages24
JournalGeometry and Topology
StatePublished - 2004

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


  • Contact structures
  • Dehn surgery
  • Floer homology
  • Seifert genus
  • Thurston norm


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