Heegaard Floer homology and alternating knots

Abstract

In an earlier paper, we introduced a knot invariant for a null-homologous knot K in an oriented three-manifold Y, which is closely related to the Heegaard Floer homology of Y. In this paper we investigate some properties of these knot homology groups for knots in the three-sphere. We give a combinatorial description for the generators of the chain complex and their gradings. With the help of this description, we determine the knot homology for alternating knots, showing that in this special case, it depends only on the signature and the Alexander polynomial of the knot (generalizing a result of Rasmussen for two-bridge knots). Applications include new restrictions on the Alexander polynomial of alternating knots.

Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper6.abs.html

References in corpus (1)

• Heegaard Floer homologies and contact structures

Cited by in corpus (116)

• Floer homology and knot complements
• Knot Floer homology and the four-ball genus
• Holomorphic disks and genus bounds
• Holomorphic disks and link invariants
• Concordance homomorphisms from knot Floer homology
• On the Khovanov and knot Floer homologies of quasi-alternating links
• Bounds for the Thurston-Bennequin number from Floer homology
• Knot Floer homology of Whitehead doubles
• Heegaard Floer homologies and contact structures
• Cables of thin knots and bordered Heegaard Floer homology
• Involutive Heegaard Floer homology
• Khovanov homology and the slice genus
• Some differentials on Khovanov-Rozansky homology
• The decategorification of sutured Floer homology
• Heegaard Floer homology and genus one, one boundary component open books
• Moving basepoints and the induced automorphisms of link Floer homology
• A spanning tree model for the Heegaard Floer homology of a branched double-cover
• Alternating sum formulae for the determinant and other link invariants
• A combinatorial spanning tree model for knot Floer homology
• Rasmussen's spectral sequences and the sl(N)-concordance invariants
• The Superpolynomial for Knot Homologies
• On knot Floer homology in double branched covers
• Longitude Floer homology and the Whitehead double
• A concordance invariant from the Floer homology of +/- 1 surgeries
• On knot Floer homology and cabling
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• Ozsvath-Szabo and Rasmussen invariants of cable knots
• Framed instanton homology and concordance
• On knot Floer width and Turaev genus
• Knot traces and concordance
• A splitting theorem for the Seiberg-Witten invariant of a homology $S^1\times S^3$
• Computing Knot Floer Homology in Cyclic Branched Covers
• A cube of resolutions for knot Floer homology
• Distinguishing topologically and smoothly doubly slice knots
• On sutured Floer homology and the equivalence of Seifert surfaces
• Dehn surgeries and rational homology balls
• Computations of Heegaard-Floer knot homology
• Cobordism invariants from BPS q-series
• Applications of involutive Heegaard Floer homology
• Knot Floer homology in cyclic branched covers
• Smooth concordance of links topologically concordant to the Hopf link
• Dehn Twists in Heegaard Floer Homology
• sl3-foam homology calculations
• Floer homology and singular knots
• Khovanov-Seidel quiver algebras and Ozsváth-Szabó's bordered theory
• Knot Floer homology, genus bounds, and mutation
• On the equivalence of contact invariants in sutured Floer homology theories
• A concordance invariant from the Floer homology of double branched covers
• A combinatorial description of knot Floer homology
• The surgery exact triangle in Pin(2)-monopole Floer homology
• Holomorphic disks and knot invariants
• Peculiar modules for 4-ended tangles
• Khovanov homology and the cinquefoil
• Invariants for Turaev genus one links
• Topologically slice knots that are not smoothly slice in any definite 4-manifold
• The Dehn surgery characterization of the trefoil and the figure eight knot
• Floer homology and embedded bipartite graphs
• Purely cosmetic surgeries and pretzel knots
• Algebras with matchings and knot Floer homology
• Contact structures and reducible surgeries
• Characterizing slopes for $5_2$
• The Ozsv{á}th-Szab{ó} and Rasmussen concordance invariants are not equal
• A note on knot Floer homology of links
• Knot Floer homology of (1,1)-knots
• Extremal Khovanov homology of Turaev genus one links
• Heegaard Floer homology and Morse surgery
• Knot Floer homology detects genus-one fibred knots
• Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree
• Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot
• Link Floer homology and the Thurston norm
• Knot Floer homology and Seifert surfaces
• Constrained knots in lens spaces
• Thin links and Conway spheres
• Embedding spheres in knot traces
• Rank Inequalities on Knot Floer Homology of Periodic Knots
• On sums of torus knots concordant to alternating knots
• Transverse Knots Distinguished by Knot Floer Homology
• Kauffman's clock lattice as a graph of perfect matchings: a formula for its height
• Cyclotomic expansion and volume conjecture for superpolynomials of colored HOMFLY-PT homology and colored Kauffman homology
• On constraints for knots to admit chirally cosmetic surgeries and their calculations
• Manolescu correction terms and knots in the three-sphere
• Upsilon type concordance invariants
• Legendrian Non-Simple Two-Bridge Knots
• On Coxeter mapping classes and fibered alternating links
• Steenrod square for virtual links toward Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links
• Heegaard-Floer homology and string links
• Filtration of Heegaard Floer homology and gluing formulas
• Heegaard Floer homologies for (+1) surgeries on torus knots
• On independence of iterated Whitehead doubles in the knot concordance group
• Symmetries and adjunction inequalities for knot Floer homology
• Knot surgery formulae for instanton Floer homology II: applications
• A Simplification of Combinatorial Link Floer Homology
• Framed Floer Homology
• Heegaard Floer correction terms of $(+1)$-surgeries along $(2,q)$-cablings
• The Morse-Novikov number of knots under connected sum and cabling
• A knot Floer stable homotopy type
• Computations of HOMFLY homology
• Floer homology and non-fibered knot detection
• A survey on the Turaev genus of knots
• Heegaard Floer homology of mapping tori II
• On the Alexander polynomial and the signature invariant of two-bridge knots
• Turaev genus, knot signature, and the knot homology concordance invariants
• On knot Floer homology for some fibered knots
• On knots with infinite smooth concordance order
• A note on the concordance $\mathbb{Z}$-genus
• A characterization of $T_{2g+1,2}$ among alternating knots
• Graph Homologies and Functoriality
• An unoriented skein exact triangle for knot Floer homology
• A note on knot Floer homology of satellite knots with (1,1)-patterns
• Alexander-Conway Polynomial State Model and Link Homology
• On the Turaev genus of torus knots
• A note on thickness of knots
• A non-torus positive 2-bridge knot does not admit chirally cosmetic surgeries
• Nontrivial Alexander polynomials of knots and links
• New family of hyperbolic knots whose Upsilon invariants are convex
• Computation of the knot Floer complex of knots of thickness one