×

An algorithm for computing some Heegaard Floer homologies. (English) Zbl 1228.57017

Heegaard Floer homology and knot Floer homology are both obtained by applying Lagrangian Floer homology to a symplectic setup associated to a Heegaard diagram. As such, the theory is defined by counting 0-dimensional components of certain moduli spaces of holomorphic disks. In the present article, the authors specify a class of Heegaard diagrams they call nice. They prove that every closed, oriented 3-manifold (with or without a knot) can be described by a nice Heegaard diagram and that for nice Heegaard diagrams the differential of both of the hat-version of Heegaard Floer homology and of the hat-version of knot Floer homology is a combinatorial object. So, there is an algorithmic way to compute the hat-version of Heegaard Floer homology of every closed, oriented 3-manifold and an algorithmic way to compute the hat-version of knot Floer homology.

MSC:

57R58 Floer homology
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] P. Ghiggini, ”Knot Floer homology detects genus-one fibred knots,” Amer. J. Math., vol. 130, iss. 5, pp. 1151-1169, 2008. · Zbl 1149.57019 · doi:10.1353/ajm.0.0016
[2] A. Juhász, ”Floer homology and surface decompositions,” Geom. Topol., vol. 12, iss. 1, pp. 299-350, 2008. · Zbl 1167.57005 · doi:10.2140/gt.2008.12.299
[3] R. Lipshitz, ”A cylindrical reformulation of Heegaard Floer homology,” Geom. Topol., vol. 10, pp. 955-1097, 2006. · Zbl 1130.57035 · doi:10.2140/gt.2006.10.955
[4] R. Lipshitz, C. Manolescu, and J. Wang, ”Combinatorial cobordism maps in hat Heegaard Floer theory,” Duke Math. J., vol. 145, iss. 2, pp. 207-247, 2008. · Zbl 1153.57027 · doi:10.1215/00127094-2008-050
[5] C. Manolescu, P. S. Ozsváth, and S. Sarkar, ”A combinatorial description of knot Floer homology,” Ann. of Math., vol. 169, pp. 633-660, 2009. · Zbl 1179.57022 · doi:10.4007/annals.2009.169.633
[6] Y. Ni, ”Knot Floer homology detects fibred knots,” Invent. Math., vol. 170, iss. 3, pp. 577-608, 2007. · Zbl 1138.57031 · doi:10.1007/s00222-007-0075-9
[7] Y. Ni, Link Floer homology detects the Thurston norm, 2006. · Zbl 1203.57005 · doi:10.2140/gt.2009.13.2991
[8] P. Ozsváth and Z. Szabó, ”Holomorphic disks and topological invariants for closed three-manifolds,” Ann. of Math., vol. 159, iss. 3, pp. 1027-1158, 2004. · Zbl 1073.57009 · doi:10.4007/annals.2004.159.1027
[9] P. Ozsváth and Z. Szabó, ”Holomorphic disks and three-manifold invariants: properties and applications,” Ann. of Math., vol. 159, iss. 3, pp. 1159-1245, 2004. · Zbl 1081.57013 · doi:10.4007/annals.2004.159.1159
[10] P. Ozsváth and Z. Szabó, ”Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary,” Adv. Math., vol. 173, iss. 2, pp. 179-261, 2003. · Zbl 1025.57016 · doi:10.1016/S0001-8708(02)00030-0
[11] P. Ozsváth and Z. Szabó, ”Holomorphic triangles and invariants for smooth four-manifolds,” Adv. Math., vol. 202, iss. 2, pp. 326-400, 2006. · Zbl 1099.53058 · doi:10.1016/j.aim.2005.03.014
[12] P. Ozsváth and Z. Szabó, ”Holomorphic disks and knot invariants,” Adv. Math., vol. 186, iss. 1, pp. 58-116, 2004. · Zbl 1062.57019 · doi:10.1016/j.aim.2003.05.001
[13] P. Ozsváth and Z. Szabó, ”Holomorphic disks and genus bounds,” Geom. Topol., vol. 8, pp. 311-334, 2004. · Zbl 1056.57020 · doi:10.2140/gt.2004.8.311
[14] P. Ozsváth and Z. Szabó, ”Knot Floer homology and the four-ball genus,” Geom. Topol., vol. 7, pp. 615-639, 2003. · Zbl 1037.57027 · doi:10.2140/gt.2003.7.615
[15] P. Ozsváth and Z. Szabó, ”Holomorphic disks, link invariants and the multi-variable Alexander polynomial,” Algebr. Geom. Topol., vol. 8, pp. 615-692, 2008. · Zbl 1144.57011 · doi:10.2140/agt.2008.8.615
[16] P. Ozsváth and Z. Szabó, ”Link Floer homology and the Thurston norm,” J. Amer. Math. Soc., vol. 21, iss. 3, pp. 671-709, 2008. · Zbl 1235.53090 · doi:10.1090/S0894-0347-08-00586-9
[17] P. Ozsváth and Z. Szabó, ”Heegaard diagrams and Floer homology,” in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1083-1099. · Zbl 1116.53062
[18] P. Ozsváth and Z. Szabó, ”Heegaard diagrams and holomorphic disks,” in Different Faces of Geometry, New York: Kluwer/Plenum, 2004, pp. 301-348. · Zbl 1091.57010
[19] J. Rasmussen, ”Floer homology and knot complements,” PhD Thesis , Harvard University, 2003.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.