A combinatorial spanning tree model for knot Floer homology

Baldwin, John A.; Levine, Adam Simon

doi:10.1016/j.aim.2012.06.006

Cited by 44 publications

(112 citation statements)

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“…Similarly, one can show that 2 w D 0 and that w 1 w 2 D w 2 w 1 for w 1 ¤ w 2 2 w f . Further degeneration arguments involving holomorphic triangles show that w commutes with the isomorphisms associated to isotopy and handleslide (cf [5,Proposition 3.6]). More transparently, w commutes with the maps associated to index 1/2 and linked index 0/3 (de)stabilization as well as the map associated to free index 0/3 (de)stabilization as long as w is not the free basepoint being added (or removed).…”

Section: Let Cfkmentioning

confidence: 99%

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On the equivalence of Legendrian and transverse invariants in knot Floer homology

2013

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Using the grid diagram formulation of knot Floer homology, Ozsváth, Szabó and Thurston defined an invariant of transverse knots in the tight contact 3-sphere. Shortly afterwards, Lisca, Ozsváth, Stipsicz and Szabó defined an invariant of transverse knots in arbitrary contact 3-manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above.

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Section: Let Cfkmentioning

confidence: 99%

“…5 Let fb 1 ; : : : ; b k g be another such basis, where each b i is obtained from a i by shifting the endpoints of a i slightly in the direction of the orientation on @S and isotoping to ensure that there is a single transverse intersection between b i and a i , as shown in Figure 4. …”

Section: The Loss Invariantsmentioning

confidence: 99%

On the equivalence of Legendrian and transverse invariants in knot Floer homology

2013

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“…also [MOST07]; (2) another by Sarkar and Wang, using nice diagrams [SW10]; (3) another by Ozsváth and Szabó, using a cube of resolutions [OS09]; (4) another by Baldwin and Levine, in terms of spanning trees [BL12]; (5) yet another recently announced by Ozsváth and Szabó, based on bordered Floer homology.…”

Section: Introductionmentioning

confidence: 99%

An introduction to knot Floer homology

Manolescu¹

2016

Physics and Mathematics of Link Homology

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Abstract. This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer homology, and the connection to three-and four-dimensional topology via surgery formulas. We also describe some conjectural relations to Khovanov-Rozansky homology.

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“…Each basepoint in a multi-pointed Heegaard diagram for (L, p) determines a differential on HFK(L, p). These basepoint actions have previously been applied in [BL12,BVVV13,Sar15,BLS,Zem16]. The combined actions, subject to anticommutation relations described in Subsection 3.3, make HFK(L, p) a Clifford module over a Clifford algebra Ω ν .…”

Section: T1mentioning

confidence: 99%

On Conway mutation and link homology

Lambert-Cole

2019

Advances in Mathematics

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We give a new, elementary proof that Khovanov homology with Z/2Z-coefficients is invariant under Conway mutation. This proof also gives a strategy to prove Baldwin and Levine's conjecture that δ-graded knot Floer homology is mutation-invariant. Using the Clifford module structure on HFK induced by basepoint maps, we carry out this strategy for mutations on a large class of tangles. Let L ′ be a link obtained from L by mutating the tangle T . Suppose some rational closure of T corresponding to the mutation is the unlink on any number of components. Then L and L ′ have isomorphic δ-graded HFK groups over Z/2Z as well as isomorphic Khovanov homology over Q. We apply these results to establish mutation-invariance for the infinite families of Kinoshita-Terasaka and Conway knots. Finally, we give sufficient conditions for a general Khovanov-Floer theory to be mutation-invariant.2010 Mathematics Subject Classification. 57M27; 57R58.The complex (CKh(D), d Kh ) possess two gradings, the quantum and homological grading, and the differential preserves the quantum grading and increases the homological grading by one. Thus the homology Kh(D) splits into the direct sum of bigraded modules Kh i,j (D) where i denotes the homological grading and j the quantum grading.Let p be a fixed basepoint in the plane contained in the diagram D. The basepoint p determines a chain map X p : CKh(D) → CKh(D) that squares to 0. The kernel of X p is a subcomplex and reduced Khovanov homology is its homology, with a shift in the quantum grading:It is an invariant of L up to isotopies supported away from p. While over Z the reduced homology depends on the component containing the basepoint, this is not true over Z/2Z.for every link L and any basepoint p. In particular, Kh(L) is well-defined independent of p.Consequently, when discussing reduced Khovanov homology over Z/2Z we will suppress any mention of the basepoint p.Khovanov homology satisfies Kunneth-type formulas for disjoint unions and connected sums. The following properties are well-known and the proofs are easy deductions from the definition of Khovanov homology and Proposition 2.1.Lemma 2.2. Let L 1 , L 2 be arbitrary links. Over Z/2Z there are isomorphismsfor any choice of connected sum.Khovanov homology satisfies an unoriented skein exact triangle. Fix a crossing of D and let D 0 , D 1 denote the 0-and 1-resolutions of D at this crossing. Since Khovanov homology is computed from a cube of resolutions, the complex CKh(D) is, up to a grading shift, the mapping cone of a chain map

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A combinatorial spanning tree model for knot Floer homology

Cited by 44 publications

References 42 publications

On the equivalence of Legendrian and transverse invariants in knot Floer homology

On the equivalence of Legendrian and transverse invariants in knot Floer homology

An introduction to knot Floer homology

On Conway mutation and link homology

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