In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.
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In this paper, we construct a canonical grading on bordered Heegaard Floer homology by homotopy classes of nonvanishing vector fields. This grading is a generalization of our construction of an absolute grading on Heegaard Floer homology and it extends the well-known grading with values in a noncommutative group define…
New signs and gradings enable detailed comparison in Heegaard Floer theory.
problem Comparing decategorified Heegaard Floer theory with modern TQFTs.
method Added signs and gradings to interval gluing theorem over Z.
result Detailed comparison possible with modern TQFTs.
We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…
We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the sp…
For knots in S^3, the bi-graded hat version of knot Floer homology is defined over Z; however, for a link L in S^3 with #|L|=l>1, there are 2^{l-1} bi-graded hat versions of link Floer homology defined over Z, the multi-graded hat version of link Floer homology is only defined over F_2 from holomorphic considerations, …
Knot Floer homology matches fixed point Floer for fibred knots.
problem Understanding the algebraic structure of knot Floer homology.
method Proved isomorphism between Knot Floer and fixed point Floer homologies.
result Knot Floer homology of fibred knots matches fixed point Floer homology.
Heegaard Floer homology, first introduced by P. Ozsvath and Z. Szabo, associates to a 3-manifold Y a family of relatively graded Abelian groups HF(Y,t), indexed by Spin^c structures t on Y. In the case that Y is a rational homology sphere, Ozsvath and Szabo lift the relative Z-grading to an absolute Q-grading. This ind…
Study shows Conway mutation preserves a specific link invariant.
problem Investigating symmetry properties of peculiar modules.
method Analysis of Heegaard Floer invariants of 4-ended tangles.
result Conway mutation preserves the hat flavor of relatively δ-graded Heegaard Floer theory of links.
The paper studies knot Floer homology under Murasugi sum and establishes graded isomorphisms.
problem Behavior of knot Floer homology under Murasugi sum.
method Established a graded version of Ni's isomorphism and proved τ=g for each summand.
result Graded isomorphisms between extremal knot Floer homologies of Murasugi sum and tensor products.
In joint work with Yang Huang, we defined a canonical absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields. A similar grading was defined on embedded contact homology by Michael Hutchings. In this paper we show that the isomorphism between these homology theories defined by Colin-G…
Real Heegaard Floer homology gets a new grading for certain 3-manifolds.
problem Real Heegaard Floer homology groups get an absolute Z/2 grading under specific conditions.
method Analyzes real Heegaard Floer homology groups with an involution and nullhomologous fixed points.
result Defines a new invariant of knots equal to the Alexander polynomial evaluated at i.
Link cobordism maps are graded and determine surface genus.
problem Understanding graded structures in link Floer homology.
method Grading change formula and cobordism maps.
result Link cobordism maps are determined by surface genus.
The pong algebramath.GT
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) inter…
Unified model for knot polynomials using quantum Heegaard diagrams.
problem Categorify knot polynomials using Floer homology.
method Construct quantum Heegaard diagrams, identify gradings, and define a two-variable graded intersection.
result Unified intersection model recovers Alexander and Jones polynomials.
New generalization of knot Floer homology rank conjecture.
problem Rank of knot Floer homology in the next-to-top Alexander grading.
method Independent generalization to a family of knots with specific conditions.
result Generalization verified for a family of knots with Seifert surfaces satisfying certain conditions.
Concordance maps in math.GT
A new map from knot concordance to Heegaard Floer homology.
problem Understanding how knot concordance affects Heegaard Floer homology.
method Using sutured Floer homology, we define a module homomorphism that preserves gradings.
result The constructed map respects Alexander and Maslov gradings.
Study shows non-trivial knot Floer homology for specific knots.
problem Proving non-triviality of knot Floer homology for certain knots.
method Analyzing Alexander grading and proving non-triviality in specific knots.
result Non-trivial knot Floer homology for certain knots, answering a question posed by Baldwin and Vela-Vick.
Generators and relations for knot Floer homology algebras computed.
problem Computing knot Floer homology using algebras defined by Ozsváth and Szabó.
method Generators and relations description, homology computation, formality determination.
result Generators and relations for the algebras are found, and their homology is computed.
We develop a method of calculation for the symplectic Floer homology of composite knots. The symplectic Floer homology of knots defined in \cite{li} naturally admits an integer graded lifting, and it formulates a filtration and induced spectral sequence. Such a spectral sequence converges to the symplectic homology of …
Given a diagram of a link K in S^3, we write down a Heegaard diagram for the branched-double cover Sigma(K). The generators of the associated Heegaard Floer chain complex correspond to Kauffman states of the link diagram. Using this model we make some computations of the homology \hat{HF}(Sigma(K)) as a graded group. W…
Researchers prove skein relations for tangle Floer homology.
problem Understanding tangle Floer homology through skein relations.
method Combinatorial proof of skein relations for tangle Floer homology.
result Tangle Floer homology satisfies skein relations.
The paper develops a new Floer theory for 3-manifolds with involutions.
problem Constructing a Floer theory for 3-manifolds with symmetries.
method Develops a framed real monopole Floer homology for 3-manifolds with involutions and provides definitions for invariants.
result Defines -valued framed real Seiberg--Witten invariants for 4-manifolds with involutions.
An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fieldsmath.SG
For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…
New connection between dynamics and Heegaard Floer homology.
problem Understanding pseudo-Anosov flows and their dynamics.
method Using Heegaard Floer homology and veering branched surfaces, the paper constructs a chain complex to categorify the zeta function of a pseudo-Anosov flow.
result Generators of the chain complex correspond to closed multi-orbits of the flow, and their homology classes have dynamical significance.
Study calculates instanton Floer homology for surgeries on L-space knots.
problem Computing the framed instanton Floer homology of surgeries on L-space knots.
method Used Baldwin-Sivek contact invariant and proved homogeneity of the invariant.
result Framed instanton Floer homology agrees with Heegaard Floer homology for L-space knots.
A Categorification of the HOMFLY-PT Polynomial with a Spectral Sequence to Knot Floer Homologymath.GT
Spectral sequence links knot Floer homology to HOMFLY-PT polynomial.
problem Connecting knot Floer homology to HOMFLY-PT polynomial.
method Constructing a spectral sequence from cube of resolutions.
result Spectral sequence converges to knot Floer homology and ranks HOMFLY-PT homology.
Bordered Heegaard Floer homology is an invariant for 3-manifolds, which associates to a surface F an algebra A(F), and to a 3-manifold Y with boundary, together with an orientation-preserving diffeomorphism from F to the boundary of Y, a module over A(F). In a previous paper, we defined relative Z/2 differential gradin…
Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary …
Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.
problem Understanding gradings in annular Khovanov homology.
method Infinite families of annular links, computations, satellite operations, link splitting spectral sequence.
result Existence of links with unbounded annular Khovanov gradings but bounded Floer gradings.
Formula refines knot Floer homology via surgery on 3-manifolds.
problem Computing knot Floer homology after surgery.
method Develops a surgery formula for knot Floer homology using a dual knot.
result Derives a combinatorial formula for the Alexander grading.
Study shows knot grading properties in specific spaces.
problem Understanding knot grading in specific geometric spaces.
method Analyzes knot Floer homology in generalized L-spaces with low first Betti number.
result Rank inequality for knot Floer homology supports single grading.
The singular instanton Floer homology was defined by Kronheimer and Mrowka in connection with their proof that the Khovanov homology is an unknot detector. We study this theory for knots and two-component links using equivariant gauge theory on their double branched covers. We show that the special generator in the sin…
New algebras model Ozsváth-Szabó's Kauffman-states.
problem Modeling Ozsváth-Szabó's Kauffman-states.
method Defined new differential graded algebras A(n,k,S) and showed quasi-isomorphism to B(n,k,S).
result Natural emergence of Ozsváth-Szabó's gradings.
Guts of nearly fibered knotsmath.GT
New models found for guts of nearly fibered knots.
problem Characterizing nearly fibered knots topologically.
method Provided three models for the guts of nearly fibered knots.
result Nearly fibered condition can be purely topologically characterized.
Heegaard Floer theory is a kind of topological quantum field theory, assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented 4-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with ext…
New theorem connects tangle complements in 3-manifolds via Floer homology.
problem Understanding tangle complements in arbitrary 3-manifolds.
method Adapting holomorphic polygon counts to bordered-sutured Floer homology.
result Exact triangle relation for tangle complements in arbitrary 3-manifolds.
This paper uses Heegaard Floer theory to study pseudo-Anosov flows and their periodic points.
problem Understanding the differential of Heegaard Floer chain complexes associated with pseudo-Anosov flows.
method Introduces a refined grading to analyze the homology of subcomplexes representing irreducible multi-orbits.
result The homology of these subcomplexes is 1-dimensional, providing insights into periodic points of pseudo-Anosov flows.
We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule define…
Study links with annuli using sutured Floer homology.
problem Characterize links with specific cable structures.
method Apply sutured Floer homology techniques.
result Characterizations of links with -cables and -cables.
The paper extends a knot invariant to graphs and connects it to homology cylinders.
problem Understanding the structure of homology cobordism groups.
method Using tangle Floer homology, the authors define a new invariant for embedded graphs and prove a concatenation formula.
result The new invariant induces a homomorphism on the homology cobordism group of homology cylinders.
Paper generalizes sutured Floer homologies with new algorithms and polytopes.
problem Computing and understanding sutured Floer homologies for complex manifolds.
method Grading shifting property, algorithms, polytopes, canonical decompositions.
result Established new bounds and detection results for sutured Floer homologies.
Proves nontrivial knot Floer homology for fibered knots.
problem Determining the knot Floer homology of fibered knots.
method Uses Alexander grading and 3-manifold surgeries.
result Knot Floer homology is nontrivial in the next-to-top Alexander grading for fibered knots.
In this article we study the Heegaard Floer link homology of -torus links. The Alexander multigradings which support non-trivial homology form a string of unit hypercubes in , and we compute the ranks and gradings of the homology in nearly all Alexander gradings. We also conjecture a compl…
Algorithm computes meridional knot summands in Dehn surgeries.
problem Computing meridional summands in Dehn surgeries for knots.
method Introduced bordered Floer bimodule algorithm for knot complements and meridians.
result Grading of the module computes meridional knot summands in Dehn surgeries.
To a link L in the 3-sphere, we associate a spectral sequence whose E^2 page is the reduced Khovanov homology of L and which converges to a version of the monopole Floer homology of the branched double cover. The pages E^k for k > 1 depend only on the mutation equivalence class of L. We define a mod 2 grading on the sp…
Study shows how concordance surgery impacts a 4D knot invariant.
problem Understanding how concordance surgery affects a specific 4D knot invariant.
method Used sutured Floer TQFT and a perturbed version of sutured Floer homology.
result Formula involving the graded Lefschetz number of the concordance map on knot Floer homology.