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.
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.
Khovanov homology identifies a specific knot.
problem Identifying knots using Khovanov homology.
method Comparing Khovanov homology of links to that of a specific knot.
result Khovanov homology can distinguish between links and the specific knot T(2,6). Explains Khovanov homology and its applications.
problem Understanding Khovanov homology and its applications.
method Expository lecture notes covering Jones polynomial, Khovanov homology, cobordism category, spectral sequences, and skein lasagna modules.
result Explains the Jones polynomial, Khovanov homology, and their applications.
New link detection results using closures of 3-braids.
problem Link detection using homology theories.
method Closure operations on 3-braids and homology theories.
result Detection of specific links using link Floer homology, Khovanov homology, and annular Khovanov homology.
Paper constructs a spectral sequence linking annular Khovanov homology to reduced Khovanov homology.
problem Classifying links with minimal annular Khovanov homology rank.
method Constructs a spectral sequence converging from annular Khovanov homology to reduced Khovanov homology.
result Classifies links with minimal rank of annular Khovanov homology.
We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homolog…
Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in Z2 generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras…
Detects (2,5) torus knot using Khovanov homology and Floer homology.
problem Detecting the (2,5) torus knot using Khovanov homology.
method Combines Floer homology, Khovanov homology, and surface homeomorphisms.
result Proves Khovanov homology detects the (2,5) torus knot.
New stable homotopy refinement of quantum annular Khovanov homology.
problem Quantum topological Hochschild homology and annular Khovanov spectra.
method Introducing quantum topological Hochschild homology (qTHH) and constructing a new stable homotopy refinement of quantum annular Khovanov homology.
result The new stable homotopy refinement agrees with qTHH of spectral Chen-Khovanov tangle bimodules and recovers earlier work.
Study shows specific states produce Khovanov homology torsion.
problem Understanding torsion in Khovanov homology.
method Proved specific sum of enhanced states produce torsion.
result Proved specific states produce Khovanov homology torsion of order two.
Knots without 2-torsion have minimal Khovanov homology rank.
problem Proving the minimality of Khovanov homology rank for knots without 2-torsion.
method Analyzing the effect of rational tangle replacements on Khovanov homology rank.
result Knots without 2-torsion have minimal Khovanov homology rank.
New mathematical tools for studying knots and links.
problem Understanding knot and link diagrams using topological invariants.
method Introducing Khovanov Laplacian and Khovanov Dirac to study diagrams.
result The harmonic spectrum retains Khovanov homology invariants, while non-harmonic spectra reveal additional information.
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 method computes first Vassiliev derivative of Khovanov homology.
problem Computing Vassiliev derivatives of Khovanov homology.
method Developed a crux complex to compute the first derivative.
result Direct computation of the first derivative of Khovanov homology.
Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to…
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.
problem Understanding the relationship between Khovanov homology and decomposable Lagrangian cobordisms.
method Utilized previously defined filtered invariants to give obstructions.
result Partial answer to Ekholm, Honda, and Kálmán's question about Khovanov homology and decomposable Lagrangian cobordisms.
Khovanov-Rozansky homology shows periodic links have group actions.
problem Understanding periodic links through homology.
method Showed Khovanov-Rozansky slN-homology has group actions on m-periodic links. result Proved an analog of periodicity criterion using slN-homology. We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a (2,n)-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov homology. In the classical case of Khovanov homology we prove the concrete connectio…
We define a link homology theory that is readily seen to be both isomorphic to reduced odd Khovanov homology and fully determined by data impervious to Conway mutation. This gives an elementary proof that odd Khovanov homology is mutation invariant, and therefore that mod 2 Khovanov homology is mutation invariant. We a…
In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain Z2 torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a…
Proves a categorified relation in Khovanov homology.
problem None explicitly stated; focuses on categorification of a relation.
method Categorified analogue of Kontsevich's 4T relation on Khovanov homology.
result Proof of categorified 4T relation in Khovanov homology.
New Khovanov homology for links with multiple punctures.
problem Defining a new Khovanov homology for links with multiple punctures.
method Defined a variant of Khovanov homology for links in thickened disks with multiple punctures, related to previous work by spectral sequences.
result Spectral sequences recover annular Khovanov homology to Khovanov homology.
Khovanov homology detects specific links.
problem Detecting specific links using Khovanov homology.
method Using Z/2-coefficients, proving detection of specific links.
result Khovanov homology detects L7n1 and a trefoil with meridian union.
New equivariant version of Khovanov homology for annuli.
problem Developing a new mathematical framework for annular Khovanov homology.
method Using Frobenius algebra and equivariant cohomology of CP1. result Introduced an equivariant version of the Temperley-Lieb algebra.
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. Extends Khovanov bracket to link cobordisms, proving functoriality up to scalars.
problem Proving functoriality of Khovanov homology under link cobordisms.
method Extending generalized Khovanov bracket to smooth link cobordisms in R^3×I and proving functoriality up to global invertible scalars.
result Generalized Khovanov bracket is functorial up to global invertible scalars.
Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain …
Adapts scanning algorithm for odd Khovanov homology.
problem Computing odd Khovanov homology efficiently.
method Uses mapping cone construction instead of tensor product.
result Determines odd Khovanov homology of 3-strand torus links.
Study shows infinite hyperbolic knots with odd torsion in Khovanov homology.
problem Understanding torsion in Khovanov homology for hyperbolic knots.
method Analyzes knots in S3 with specific torsion in Khovanov homology. result Infinitely many hyperbolic knots with odd torsion in Khovanov homology.
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.
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…
Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khova…
New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.
problem Distinguishing exotic surfaces in the 4-ball that are not diffeomorphic.
method Developed new techniques for distinguishing cobordism maps on Khovanov homology using knot symmetries and braid factorizations.
result Distinguishes smooth surfaces in the 4-ball that are exotically knotted.
Classifies links with small Khovanov homology ranks.
problem Classifying links with specific ranks in Khovanov homology.
method Previous results combined with new classifications.
result All links with ranks ≤ 8 and three-component links with ranks ≤ 12 are classified.
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 operators in Khovanov-Rozansky homology exhibit symmetry.
problem Symmetry in Khovanov-Rozansky homology.
method Defined new commuting operators Fk and proved F2 satisfies hard Lefschetz property. result Symmetry in Khovanov-Rozansky homology is confirmed.
In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an…
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
In this paper, we study the Khovanov homology of cable links. We first estimate the maximal homological degree term of the Khovanov homology of the (2k+1, (2k+1)n)-torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the (2k+1, (2k+1)n)-torus li…
Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this …
Study supports conjecture about pretzel links' homology.
problem Investigate extreme Khovanov homology of pretzel links.
method Geometric realizations of the independence simplicial complex.
result Supports conjecture that extreme Khovanov homology is torsion-free.
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded vector space that vanishes if and only if the strongly invertible knot is trivial.…
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.
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.
Ozsváth, Rasmussen and Szabó constructed odd Khovanov homology. It is a link invariant which has the same reduction modulo 2 as (even) Khovanov homology. Szabó introduced a spectral sequence with mod 2 coefficients from mod 2 Khovanov homology to another link homology. He got his spectral sequence from a chain complex …