A combinatorial description of knot Floer homology

By Ciprian Manolescu, Peter S. Ozsv'ath, Sucharit Sarkar. 2006

248 citations recorded by the source corpus.

Given a grid presentation of a knot (or link) K in the three-sphere, we describe a Heegaard diagram for the knot complement in which the Heegaard surface is a torus and all elementary domains are squares. Using this diagram, we obtain a purely combinatorial description of the knot Floer homology of K.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/bbac3ab128edf569af4551100eb9ed9de862d703

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Citations in the corpus

An algorithm for computing some Heegaard Floer homologies (2006, result): “It will be interesting to compare our result with the recent work of Ciprian Manolescu, Peter Ozsváth and the first author in [5], where they gave a combinatorial description of knot Floer homology of knots in S(3), in all versions.”
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.”
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,”
Heegaard Floer homology and integer surgeries on links (2010, unknown): “We focus on the description in [9], in terms of grid diagrams, which has the advantage that it extends to all versions of link Floer homology for links in S3.”
On knot Floer width and Turaev genus (2007, unknown): “G T ] 5 S ep 2”
On the geography and botany of knot Floer homology (2014, background): “They were originally defined using ideas from symplectic geometry—specifically, the theory of pseudo-holomorphic curves— but have since been shown to admit a combinatorial definition [53,54].”
Cables of thin knots and bordered Heegaard Floer homology (2009, methodology): “Later, combinatorial versions appeared, including a method using grid diagrams [6].”
Unoriented knot Floer homology and the unoriented four-ball genus (2015, methodology): “We could see it as a modification of the construction of knot Floer homology, as defined using pseudoholomorphic curves; or alternatively, we can define it using grid diagrams as in [13, 14].”
Grid Homology for Knots and Links (2015, methodology): “Our discussion of grid homology follows the account from [136] (see also [135]), with further topics coming from [66, 85, 157, 167, 185, 204, 231].”
A cube of resolutions for knot Floer homology (2007, background): “This description leads to a concrete calculation of knot Floer homology, which is both quite different in character, and also logically independent, from the ones from [7] and [17].”
Moving basepoints and the induced automorphisms of link Floer homology (2011, methodology): “Following [7], we will represent such links by a special type of Heegaard diagrams called grid diagrams.”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “We will use the more general set-up from [MOS09], allowing multiple basepoints.”
A survey of Heegaard Floer homology (2013, methodology): “One notable result is due to Sarkar [87], which is the second completely combinatorial proof of the Milnor conjecture on the slice genera of torus knots (following Rasmussen’s proof [85] via Khovanov homology).”
Computations of Heegaard-Floer knot homology (2006, methodology): “The algorithm in [16] starts with the observation that an arc presentation ofK ⊂ S(3)”
Involutive Heegaard Floer homology (2015, methodology): “In fact, it should be noted that ιK is in principle computable for all knots in S 3, using grid diagrams and the maps on grid complexes [28, 29].”