An algorithm for computing some Heegaard Floer homologies

By Sucharit Sarkar, Jiajun Wang. 2006

152 citations recorded by the source corpus.

In this paper, we give an algorithm to compute the hat version of Heegaard Floer homology of a closed oriented three-manifold. This method also allows us to compute the filtration coming from a null-homologous link in a three-manifold.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/5aad092323e47cf2dc9017095d7c49502e5214ef

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Citations in the corpus

Grid Homology for Knots and Links (2015, methodology): “In 2006, Sucharit Sarkar found a method that rendered some of the pseudoholomorphic disk counts in the Heegaard Floer differential combinatorial; this was further pursued in a joint paper with Jiajun Wang [206].”
The contact invariant in sutured Floer homology (2007, unknown): “The α-curves are αi = ∂(ai ×[ −1, 1]).) To determine whether EH �= 0, we apply the Sarkar–Wang algorithm [ 30 ] (also see [29]) to isotop βi so that the only regions of Σ − i αi − i βi which do not intersect the suture are quadrilaterals.”
Heegaard Floer homology and integer surgeries on links (2010, unknown): “Indeed, there exist several combinatorial descriptions of knot Floer homology, see [9, 27, 13].”
A survey of Heegaard Floer homology (2013, methodology): “The breakthrough results in this direction are due to Sarkar and Wang [89], who gave an algorithm for computing y HF pY q for an arbitrary closed, connected, oriented 3manifold Y , and to Manolescu, Ozsváth, and Sarkar [57], who gave a combinatorial characterization of knot Floer homology { HFKpS(3)”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “ now, there are several such constructions in the literature: (1) one by Manolescu-Ozsva´th-Sarkar, using grid diagrams [MOS09]; cf. also [MOST07]; (2) another by Sarkar and Wang, using nice diagrams [SW10]; (3) another by Ozsvath and Szabo´, using a cube of resolutions [OS09]; (4) another by Baldwi”
On the geography and botany of knot Floer homology (2014, methodology): “This is provided by the Sarkar-Wang algorithm for computing the hat Floer homology groups of an arbitrary 3-manifold, as well as the filtered homotopy type of the filtration of the hat complex induced by an arbitrary knot therein [85].”
Khovanov module and the detection of unlinks (2012, background): “s branched double cover. This latter invariant is deﬁned analytically, by counting holomorphic curves in the symmetric product of a Riemann surface (though recent combinatorial interpretations abound [21, 37, 19]). Since their work, this spectral sequence has had a number of generalizations [35, 10,”
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology (2007, methodology): “2 of [SW06], the boundary operator counts only Whitney disks whose domains are embedded 4–sided polygons (i.”
Bordered Heegaard Floer homology (2008, methodology): “The idea to use the Sarkar-Wang algorithm to prove a gluing result is partly inspired by work of Juhász [15].”
Heegaard Floer homology as morphism spaces (2010, methodology): “This is a nice diagram [19], so its holomorphic disks can be understood explicitly; moreover, it has a combinatorial nature which is compatible with the algebraic structure.”
A combinatorial spanning tree model for knot Floer homology (2011, methodology): “In 2006, Manolescu–Ozsváth–Sarkar [29] and Sarkar–Wang [49] discovered algorithms for computing knot Floer homology via Heegaard diagrams in which the counts of holomorphic disks are completely combinatorial.”
On the spectral sequence from Khovanov homology to Heegaard Floer homology (2008, background): “Although the definition of K̃h(L) is intrinsically combinatorial and there is now a combinatorial way to compute ĤF (−Σ(L)) [25], the higher terms in this spectral sequence have remained largely mysterious.”