On combinatorial link Floer homology

By Ciprian Manolescu, Peter S. Ozsv'ath, Zolt'an Szab'o, Dylan Thurston. 2006

177 citations recorded by the source corpus.

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients.

Subfields: Computational Mechanics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e4b4aa5e0cf7a69acbe2cd67b48da8594e8eed00

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Citations in the corpus

On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, background): “x/ is equal, up to an overall shift, to the sum of the winding numbers of U around the components of x (cf Manolescu, Ozsváth and Sarkar [23], and Manolescu, Ozsváth, Szabó and Thurston [24]) in the grid.”
Spanning trees and Khovanov homology (2006, background): “ly interesting to compare with Heegaard-Floer homology for knots. The complex in [9] has generators that correspond to spanning trees, but no combinatorial diﬀerential is known. The recent complex in [7] is completely combinatorial but has many more generators, so it is quasi-isomorphic (and possibl”
Transverse knots distinguished by knot Floer homology (2007, methodology): “Recent techniques have rendered the calculations of these groups purely combinatorial [17, 18, 33].”
Grid Homology for Knots and Links (2015, methodology): “As a by-product, the invariance proof presented here is somewhat simpler than the proof from [136].”
Heegard Floer invariants of Legendrian knots in contact three-manifolds (2008, background): “The reformulation of knot Floer homology (introduced originally in [35, 39]) through grid diagrams [26, 27] provided not only a combinatorial way of computing the knot Floer homology groups of knots in S3, but also showed a natural way of defining invariants of Legendrian and transverse knots in the”
Heegaard Floer homology and integer surgeries on links (2010, unknown): “Following the notation of [10], we use X and O markings to play the role of the basepoints zi and wi, respectively.”
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].”
AN INTRODUCTION TO KNOT FLOER HOMOLOGY (2014, unknown): “binatorial deﬁnitions—at least for the case of knots in S3. By 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 Sza”
Unoriented knot Floer homology and the unoriented four-ball genus (2015, methodology): “As verified in [14], these quantities are independent from the choice of the fundamental domain and are functions of the grid states.”
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].”
On the Khovanov and knot Floer homologies of quasi-alternating links (2007, unknown): “G T ] 2 6 M ar 2 00”
Bordered Heegaard Floer homology (2008, methodology): “The actual algebra we use is an appropriate subalgebra of A(n, k).”
Legendrian knots, transverse knots and combinatorial Floer homology (2006, methodology): “Z[U 1,...,Uℓ] HL−(L) = M d∈Z s∈Zℓ HL− d (L,s), where Ui acts as an endomorphism which is homogeneous of degree −2 for d, degree −1 for the ith component of s, and degree 0 otherwise. This R-module is [10] an oriented link invariant which, when specialized to coeﬃcients in F 2, agrees with link Floer”
Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology (2007, methodology): “in the universal cover of L(p,q). One easily constructs such a Heegaard diagram which, in fact, is a grid diagram for Kein S3 in the traditional sense [MOS06]. A simple formula for the Maslov grading [MOST06] in the cover, coupled with the relative Maslov index formula of [LL06] and a calculation of”
A combinatorial spanning tree model for knot Floer homology (2011, background): “ints and the markings. These marking changes induce isomorphisms, by Lemma 3.3. Moreover, standard arguments 2When His a grid diagram for a link L in S3, CFK(] H) is just the complex gCL(H) deﬁned in [27]. A COMBINATORIAL SPANNING TREE MODEL FOR HFK 13[ show that these Heegaard moves induce quasi-is”
A combinatorial description of knot Floer homology (2006, background): “roperty that for any generator x ∈ X, A(x) = A′(x) +κ. The proof that κ= 0 is elementary, albeit tedious. We sketch it here, leaving the details as an exercise for the interested reader; compare also [16]. One ﬁrst checks that κis a knot invariant, by verifying that it is unchanged by vertical and h”
Contact surgeries and the transverse invariant in knot Floer homology (2008, background): “The motivation for the Legendrian and transverse invariants comes from the construction in [19], which gives an invariant for Legendrian and transverse knots in the standard contact three-sphere, taking values in the combinatorial knot Floer homology of [13, 14]; see also [9, 24] for other construct”
Moving basepoints and the induced automorphisms of link Floer homology (2011, background): “All these snail domains are very similar to the snail domains that appear in [8]; following their analysis, it is not very hard to see that each of the stabilization maps, and in particular the above one, is a chain map.”