2003/11/30 by Peter Ozsvath, Zoltan Szabo · 9 citations
Mathematics · #Advanced Combinatorial Mathematics #Floer homology #Geometric and Algebraic Topology #Holomorphic function #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Magnetic monopole #Mathematical proof #math.GT #math.SG #msc:53D40 #msc:57M27 #msc:57N10 #msc:57R58
paper · pdf · doi:10.2140/gt.2004.8.311
published as Geom. Topol. 8 (2004) 311-334 · Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol8/paper8.abs.html
openalex publication_date 2004/02/14 · arxiv created 2004/03/03 · arxiv updated 2014/11/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.