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.
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fix…
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.
This paper realises the Khovanov homology of a link in the 3-sphere as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k of characteristic zero. Here we prove th…
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.
Math verifies Aganagic's proposal for Khovanov homology.
problem Recovering Khovanov homology from braid group action on Fukaya-Seidel categories.
method Direct calculation of containment between combinatorially defined braid group actions.
result Mathematical verification of Aganagic's proposal for Khovanov homology.
We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial. Consequently, we prove that the Maslov grading on the Seidel-Smith symplectic link invari…
Khovanov homology for links in S^3 via 1-tangle diagrams in annulus.
problem Computing Khovanov homology for links in S^3.
method Constructing a chain complex from a 1-tangle diagram in the annulus, using a cube of resolutions.
result A spectral sequence converging to reduced Khovanov homology.
Paper proves a symplectic inequality using trisections and contact geometry.
problem Proving a symplectic inequality for 4-manifolds.
method Used contact geometry and trisections to reduce to slice-Bennequin inequality, proving with Khovanov homology.
result Gauge-theory-free proofs of landmark 4-manifold topology results.
This paper explores adiabatic solutions of Haydys-Witten equations for knot homology.
problem Investigating instanton Floer homology and its relation to Khovanov homology.
method Analyzes decoupled Haydys-Witten equations and their equivalence to EBE solutions.
result Proposes an equivalence between adiabatic solutions of decoupled Haydys-Witten equations and non-vertical paths in EBE moduli space.
In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial differential equations in 4 or 5 dimensions. The second lecture is devoted to a descri…
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented 3-manifold Y via the symplectic geometry of so-called traceless SU(2)-character varieties. This invariant, SI(Y), which we refer to as the symplectic instanton homology of Y, was also shown…
Seidel-Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith-type inequalities. Similar-looking spectral sequences have been defined by Lee, Bar-Natan, Ozsváth-Szabó, Lipshitz-Tre…
The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.
problem Constructing reduced Khovanov homology for links in lens spaces.
method Generalizing a symplectic interpretation of reduced Khovanov homology for links in S3 and constructing cochain complexes for links in S3 and S2imesS1. result The cohomology of the constructed cochain complex for links in S2imesS1 may be a link invariant. 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.
There are some similarities between cohomology of SU(2)-representation varieties of the fundamental group of some link complements and the Khovanov homology of the links. We start here a program to explain a possible source of these similarities. We introduce a symplectic manifold M with an action of the b…
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.
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conject…
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. Paper categorifies Vassiliev skein relation for Khovanov homology.
problem Clarifying the relation between Vassiliev invariants and Khovanov homology.
method Developed a categorified version of Vassiliev skein relation on Khovanov homology.
result Khovanov homology's genus-one operation leads to a crossing change, enabling invariance under Reidemeister moves and extending to singular links.
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.
Khovanov homology detects causality in spacetimes.
problem Detecting causality in spacetimes.
method Khovanov homology applied to (2+1)-dimensional spacetimes. result Khovanov homology detects causality in specific spacetimes.
Using the symplectic geometry of certain manifolds which appear naturally in Lie theory, we define an invariant which assigns a graded abelian group to an oriented link. The relevant manifolds are transverse slices to certain nilpotent orbits inside sl_{2m}, and intersections of those with regular semisimple orbits. Th…
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.
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 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. 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 …