Study calculates Heegaard Floer homology for Seifert fibered 3-manifolds.
problem Computing Heegaard Floer homology for Seifert fibered 3-manifolds.
method Combinatorial models inspired by lattice homology.
result Connected Floer homology determines local equivalence class of ι-complex. Proves a conjecture about knotted spheres using plane Floer homology.
problem Proving a conjecture about knotted spheres.
method Using Daemi's plane Floer homology as a key tool.
result Proves a conjecture regarding the odd Khovanov cobordism maps associated to knotted spheres.
We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…
Paper studies spectral invariants and monopole Floer homology for rational homology three-spheres.
problem Tackles the existence of positive scalar curvature metrics on ribbon homology cobordisms.
method Defines an R-filtration on the equivariant complex of monopole Floer homology via Chern-Simons-Dirac functional, leading to a spectral invariant.
result Shows that the spectral invariant provides an obstruction to the existence of positive scalar curvature metrics on ribbon homology cobordisms.
Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
A new model calculates colored Jones polynomials using homology.
problem Calculating colored Jones polynomials for different colors.
method Using a homological model based on the Lawrence representation and Kohno's result.
result Colored Jones polynomials are described as a graded intersection pairing of homology classes.
We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.
New algebraic method for knot Floer homology computation.
problem Computing knot Floer homology efficiently.
method Extending bordered Floer homology to partial knot projections and establishing a pairing result.
result Identification of knot Floer homology with its algebraic definition.
Bounds on homology of hyperbolic orbifolds using simplicial models.
problem Bounding homology of hyperbolic orbifolds.
method Efficient simplicial model of the thick part of hyperbolic orbifolds.
result Linear bounds on Betti numbers and torsion homology in terms of volume.
This paper shows how the Formal Knot Theory state model for the Alexander-Conway polynomial is related to Knot Floer Homology. In particular we prove a parity result about the states in this model that clarifies certain relationships of the model with Knot Floer Homology.
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the J…
Lectures on link homology and its algebraic/geometric models.
problem Defining and understanding link homology.
method Definition and properties of Khovanov-Rozansky triply graded homology, and three geometric models.
result Three geometric models for link homology: braid varieties, Hilbert schemes, and affine Springer fibers.
We define a model for the homology of manifolds and use it to describe the intersection product on the homology of compact oriented manifolds and to define homological quantum field theories which generalizes the notions of string topology introduced by Chas and Sullivan and homotopy quantum field theories introduced b…
Clarifies sign ambiguity in Khovanov homology functoriality.
problem Sign ambiguity in Khovanov homology functoriality.
method Blanchet's oriented model and cobordisms with singularities.
result Strict functoriality established for the oriented model.
This paper is dedicated to Oleg Viro on his 60-th birthday. The paper is about Khovanov homology and its relationships with statistical mechanics models such as the Ising model and the Potts model. We give a relatively self-contained introduction to Khovanov homology, and also a reformulation of the Potts model in term…
The study of higher-order homology embeddings for manifold topology.
problem Understanding the structure of higher-order homology embeddings to disclose geometric or topological information.
method Analysis of the null space of the k-th order Laplacian and proposing an algorithm to factorize the homology embedding. result The proposed spectral loop detection algorithm is more efficient and effective on various data types.
This paper proves a conjecture about knot homologies.
problem Proving a spectral sequence from reduced triply graded homology to knot Floer homology.
method Constructing a bigraded spectral sequence from gl0 homology to knot Floer homology. result Proof of the Dunfield-Gukov-Rasmussen conjecture.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
A geometric model for twisted K-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of K-homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric K-homology to the new g…
Unified theories for colored sl(2) knot homology.
problem Equivalence of different models for colored sl(2) knot homology.
method Conceptualized properties into a Chebyshev system and proved its uniqueness.
result Equivalence of Khovanov and Cooper-Krushkal models for the unknot.
We use a spanning tree model to prove a result of E. S. Lee on the support of Khovanov homology of alternating knots.
We iterate Manolescu's unoriented skein exact triangle in knot Floer homology with coefficients in the field of rational functions over Z/2Z. The result is a spectral sequence which converges to a stabilized version of delta-graded knot Floer homology. The (E2,d2) page of this spectral sequence …
We consider a homological enlargement of the mapping class group, defined by homology cylinders over a closed oriented surface (up to homology cobordism). These are important model objects in the recent Goussarov-Habiro theory of finite-type invariants of 3-manifolds. We study the structure of this group from several d…
Proves properties of instanton knot Floer homology and connected sum formula.
problem Properties of instanton knot Floer homology.
method General properties of sutured Floer theories, Heegaard Floer and monopole Floer settings.
result Connected sum formula for instanton knot Floer homology.
Paper constructs Morse-Floer homology for Dirac-geodesics.
problem Existence of Dirac-geodesics under super-quadratic perturbation.
method Morse-Floer theory applied to Dirac-geodesics with spectral sequence computation.
result Explicit computation of Dirac-geodesics' homology leading to existence results.
It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzb…
Extends Khovanov homology spectral sequence using Heegaard Floer homology.
problem Relating Khovanov homology and Heegaard Floer homology of branched double covers.
method Involutive Heegaard Floer homology, bordered Floer homology, surgery exact triangle.
result Establishes spectral sequence connecting Khovanov homology and Heegaard Floer homology.
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.
problem Extending Morse-Novikov Homology with differential graded coefficients.
method Constructs a Morse-Novikov complex and proves the existence of a Chas-Sullivan-like product for a fibration.
result Proves the existence of a Chas-Sullivan-like product on the Novikov completion of a fibration.
5D gauge theories are dual to 3D and 2D models via Floer homologies.
problem Exploring dualities in 5D gauge theories and their 3D and 2D counterparts.
method Using Landau-Ginzburg models and Floer homologies, the paper establishes dualities between different gauge theories and their associated homologies.
result Dual A∞-categories of Floer homologies are derived, proving mirror symmetry and Langlands duality. A formula for bordered Floer homology of concordances and satellites
problem Computing knot Floer homology for concordances and satellites
method Combinatorial method for bordered Floer homology
result Computation of knot Floer homology cobordism map
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a …
Refines Khovanov homology using signed Burnside categories.
problem Stable homotopy refinement of Khovanov homology.
method Signed Burnside category approach to compare Blanchet and Khovanov chain complexes.
result Stable homotopy type construction for link diagrams.
3D topological models link to HOMFLYPT homology via braids.
problem Understanding topological invariants of links and braids.
method 3D topological B-models with Hilbert schemes of points.
result Hilbert space of braids corresponds to HOMFLYPT homology.
Unified framework for proving functoriality of various link homology theories.
problem Functoriality and Reidemeister invariance of link homology theories.
method Unified framework leveraging Khovanov homology and various link homology theories.
result Stronger functoriality results by avoiding spectral sequences.
The paper uses Heegaard Floer homology to study homology cobordism groups.
problem Understanding the structure of homology cobordism groups.
method Defined an invariant using Heegaard Floer homology, analogous to Stoffregen's connected Seiberg-Witten Floer homology.
result Computed involutive correction terms for certain three-manifolds.
Khovanov and chromatic homologies show similar patterns, improving chromatic bounds and computing Jones polynomial.
problem Understanding similarities between Khovanov and chromatic homologies.
method Analyzing isomorphism and improving bounds using explicit formulas.
result Improved bounds for chromatic homology and explicit formula for chromatic homology rank.
Maps quandle homology to relative group homology.
problem Understanding the relationship between quandle and group homology.
method Introducing a chain map and constructing quandle cocycles.
result Relates quandle homology to relative group homology through triangulations.
Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
Shapley Homology measures sample influence on neural networks' manifold topology.
problem Assumption of iid samples simplifies manifold analysis in machine learning.
method Shapley Homology framework quantifies sample influence on neural networks' manifold topology.
result Higher influence scores correlate with greater impact on neural network accuracy.
The paper examines lattice homology invariants of Seifert homology spheres.
problem Understanding homology cobordism invariants for Seifert fibered integral homology 3-spheres.
method Utilizes lattice homology and Heegaard Floer homology to study invariants.
result Reproves and extends the invariance of Seifert homology spheres' d-invariants and maximal monotone subroots. Identifies a mod-p triple cup product for rational homology 3-spheres with specific first homology.
problem Locally flat embeddings in S4 for rational homology 3-spheres method Using triple torsion linking form and torsion-linking duality
result Identifies the mod-p triple cup product for specific rational homology 3-spheres Detects figure-eight knot using Khovanov homology.
problem Detecting the figure-eight knot.
method Using Dowlin's spectral sequence from Khovanov homology to knot Floer homology.
result Reduced Khovanov homology (over Q) detects the figure-eight knot.