Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.
Study detects a specific type of link using annular Khovanov homology.
problem Detecting a specific type of three-strand weaving link.
method Combines braid detection with rigidity theorem to determine (σ1σ2−1)N up to conjugacy. result Annular Khovanov homology detects the underlying unoriented annular link KN. Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.
problem Proving module structure on odd Khovanov homology.
method Geometric description of module structure and link cobordism.
result Combinatorial proof of odd invariant for ribbon 2-knots.
Khovanov homology for pro-tangles and spectral sequences
problem Developing a framework for Khovanov homology for pro-tangles and spectral sequences
method Using pro-tangles, simplicial presheaves, and spectral sequences
result Establishing a fully faithful embedding and an algebraic spectral sequence for pro-tangles
Enhanced Khovanov TQFT using basepoints for nonorientable surfaces.
problem Invariance of nonorientable surfaces in Khovanov homology.
method Basepoint actions in Khovanov homology over the field with two elements.
result Establishes invariance under connected sums with specific surfaces.
Homology of torus knots stabilizes to loop space homology.
problem Computing homology of complex Grassmannians and torus knots.
method Colored sl(N) homology and free loop space computation. result Khovanov homology of torus knots stabilizes to loop space homology.
New operations match Steenrod squares on Khovanov homology.
problem Matching Steenrod squares with operations on Khovanov homology.
method Applied cup-i products to Khovanov functor and proved agreement with Steenrod squares.
result Lipshitz-Sarkar's Sq^2 agrees with Morán's sq^2.
New link homology theories for RP3 yield distinct invariants.
problem Extending link homologies for links in RP3. method Introducing new Lee and Bar-Natan spectral sequences for links in RP3. result New Lee and Bar-Natan invariants are distinct from existing ones.
Khovanov homology extended to 3-manifolds, linking tangles.
problem Extending Khovanov homology to 3-manifolds.
method Cutting 3-manifolds into intervals, constructing type D and A structures, gluing back.
result Recovering Khovanov homology of links in 3-manifolds.
Khovanov homology invariant under Conway mutation.
problem Invariance of Khovanov homology under specific transformations.
method Strong geography restrictions and homological mirror symmetry.
result Classification of components of a Khovanov multicurve invariant.
Constructs y-ifications of Khovanov homology and proves compatibility with HOMFLY--PT.
problem Distinguishing knots with identical Khovanov and HOMFLY--PT homologies.
method Elementary construction within Bar-Natan's framework for tangles, defining e-action on y-ifications. result New structures distinguish knots with identical homologies, e.g., Conway and Kinoshita-Terasaka knots.
Khovanov homology shows (4,5) torus knot is a summand in its concordance class.
problem Understanding the global ribbon minima in knot concordance classes.
method Application of Khovanov homology to ribbon concordance.
result Khovanov homology of (4,5) torus knot is a summand in any knot's homology.
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.
We simplify Khovanov homology for torus braids using Gaussian elimination.
problem Computing Khovanov homology for torus braids is complex and computationally intensive.
method Applying Gaussian elimination to reduce the number of generators in the Khovanov chain complex.
result We provide a bound on the number of generators in the whittled complex at fixed homological degree.
Investigates polynomial time algorithms for computing Khovanov homology of braids.
problem Computing Khovanov homology for general braids is intractable.
method Examines polynomial time algorithms for 3-braids and a variation of the scanning algorithm for more general braids.
result Shows that for 3-braids, Khovanov homology can be computed in polynomial time, while for more general braids, it can be computed in polynomial time for bounded homological degrees.
Python tool calculates cobordism maps in Khovanov homology.
problem Computing cobordism maps on Khovanov homology.
method Developed a Python module to calculate these maps.
result Computed cobordism maps for all incompressible Seifert surfaces of prime knots up to 10 crossings.
New bounds on Khovanov homology for positive links families.
problem Bounding the maximum non-vanishing quantum degree of Khovanov homology.
method Diagram-independent upper bounds constructed using Khovanov homology.
result Constructs upper bounds on Khovanov homology, providing positivity obstructions.
Study symmetries in smoothed polygonal links.
problem Computing Khovanov homology and link invariants.
method Cube of resolutions with combinatorial structure.
result New group-theoretic invariants of links.
Odd Khovanov homology gets a new algebraic action from super foams.
problem Understanding the algebraic structure of odd Khovanov homology.
method Introducing a local gl1∣1-action on odd Khovanov homology via super foams. result The action of gl1∣1 on odd Khovanov homology is shown to arise from super foams. Defines a new symplectic Khovanov homology for links in fibered 3-manifolds.
problem No specific problem stated, but deals with Khovanov homology for links in fibered 3-manifolds.
method Defines a symplectic Khovanov type homology for a transverse link in a fibered closed 3-manifold with an auxiliary loop.
result Conjectural combinatorial dgas for surface categories, higher-dimensional analogs of strands algebras.
Extends odd Khovanov bracket to link cobordisms and proves functoriality up to sign.
problem Proving functoriality of odd Khovanov homology up to sign.
method Extending odd Khovanov bracket to link cobordisms and proving functoriality up to sign.
result Functoriality of odd Khovanov homology up to sign.
Extends Khovanov homology to surfaces with singularities.
problem Computing invariants for surfaces with singularities.
method Functorial extension of Khovanov homology to surfaces with double points.
result Induces a map between Khovanov homology groups of boundary links.
New method uses Khovanov homology to distinguish exotic 4-manifolds.
problem Distinguishing exotic compact, orientable 4-manifolds.
method Introduces a new, simple way of using Khovanov homology.
result First analysis-free proof of existence of exotic Mazur manifolds.
Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
Paper defines end Khovanov homology to detect exotic planes.
problem Detecting exotic Lagrangian and symplectic planes in 4-space.
method Define and apply end Khovanov homology to surfaces in 4-space.
result First known examples of exotic Lagrangian and symplectic planes found.
Khovanov homology invariant proved for links in RP3.
problem Proving Khovanov homology invariant for links in RP3. method Established functoriality of Khovanov homology for link cobordisms in IimesRP3. result Khovanov homology is an invariant of links in unparametrized RP3. Alternative proof of Khovanov's up-to-sign functoriality for odd Khovanov homology.
problem Functoriality of Khovanov's odd Khovanov homology.
method Extending Hochschild (co)homology to quasi-associative algebras and applying it to Khovanov's homology.
result First proof of functoriality of Naisse and Putyra's tangle theory up to unit.
This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
Computes Steenrod squares on Khovanov homology for knots up to 11 crossings.
problem Computing Steenrod squares on Khovanov homology.
method Spatial refinements of even and odd Khovanov homology, computation of Sq^2.
result Steenrod squares Sq^2 on Khovanov homology spaces are determined for knots up to 11 crossings.
We compute Khovanov homology for tangles using TQFT.
problem Khovanov homology for tangles is not well studied or computed.
method Topological Quantum Field Theory (TQFT) construction.
result A comprehensive method for computing Khovanov homology of tangles.
Method for computing Khovanov homology of tangles.
problem Limited explicit computational studies of Khovanov homology for tangles.
method Arc reduction approach to compute Khovanov homology.
result Derived and computed Poincaré polynomials for simple and complex tangles.
New torsion patterns found in Khovanov homology of link diagrams.
problem Identifying and characterizing torsion elements in Khovanov homology.
method Analyzing link diagrams to find new torsion patterns, using Khovanov chain complex submodules.
result Most torsion elements in the same Khovanov module are distinct.