Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.
problem Understanding transverse link invariants in the annular setting.
method Constructs a stable homotopy type for annular links and defines a map to the Khovanov skein spectrum.
result At extreme gradings, the map from the Khovanov spectrum to the Khovanov skein spectrum recovers the cohomotopy transverse invariant.
Extends quantum annular homology to infinite sets.
problem Quantum annular homology and its applications.
method Extension of Burnside categories to infinite sets and application to quantum annular Khovanov spectrum.
result Quantum annular Khovanov spectrum with infinite cyclic group action.
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 prove that the spectrum constructed by González-Meneses, Manchón and the second author is stably homotopy equivalent to the Khovanov spectrum of Lipshitz and Sarkar at its extreme quantum grading.
We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.
Proves Khovanov homology has no torsion for bipartite circle graphs.
problem Proving properties of Khovanov homology for bipartite circle graphs.
method Proved homotopy equivalence of independence complexes to wedges of spheres.
result Extreme Khovanov homology has no torsion.
Functor decomposes Khovanov spectra for non-alternating diagrams.
problem Computing Khovanov spectra for diagrams without alternating pairs.
method Functor from cube to Burnside 2-category, decomposition into simplicial complexes.
result Homotopy type of almost-extreme Khovanov spectra computed.
Given an m-periodic link L⊂S3, we show that the Khovanov spectrum XL constructed by Lipshitz and Sarkar admits a homology group action. We relate the Borel cohomology of XL to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra …
In a previous paper we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen's …
Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.
problem Understanding Jones-Wenzl projectors in Khovanov spectra.
method Constructing and studying lifted projectors via maps and polynomial actions.
result Complete computation of 3-colored Khovanov spectrum of the unknot, proving conjectures.
O. Plamenevskaya associated to each transverse knot K an element of the Khovanov homology of K. In this paper, we give two refinements of Plamenevskaya's invariant, one valued in Bar-Natan's deformation of the Khovanov complex and another as a cohomotopy element of the Khovanov spectrum. We show that the first of these…
In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinement…
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological st…
New insights into Khovanov homology complexity and topological structure.
problem Complexity of computing Khovanov homology for closed braids.
method Analysis of independence simplicial complexes and polynomial time algorithms.
result Independence simplicial complexes associated to 4-braid diagrams are homotopy equivalent to wedges of spheres.
Lifts an sl2 action to annular Khovanov homology's stable refinement.
problem Stable refinement of annular Khovanov homology's sl2 action. method Lifts actions of sl2 generators to maps of spectra, using cancellations in cube of resolutions. result Commutativity of sl2 action with Steenrod algebra action. Refines quantum annular homology using stable homotopy methods.
problem Quantum annular homology lacks a stable homotopy refinement.
method Equivariant Burnside category approach, cyclic group action.
result Stable homotopy refinement of quantum annular homology constructed.
We set up foundations of representation theory over S, the sphere spectrum, which is the `initial ring' of stable homotopy theory. In particular, we treat S-Lie algebras and their representations, characters, gln(S)-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, w…
The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.
problem Developing a stable homotopy type for planar trivalent graphs with perfect matchings.
method Defining a cover functor from the 2-factor flow category to the cube flow category, realizing the 2-factor spectrum, and showing it's an invariant.
result The stable homotopy type of the 2-factor spectrum is an invariant of planar trivalent graphs with perfect matchings.
Simplified Khovanov polynomials for bipartite links.
problem Computing Khovanov polynomials for bipartite links.
method Reduced Khovanov-Rozansky technique to Kauffman-Khovanov cycle calculus.
result Consistency demonstrated between reduced technique and bipartite Khovanov polynomials.
This paper explores Khovanov adequacy in knot theory.
problem Understanding Khovanov homology and its adequacy.
method Using independence complexes and homotopy type calculations.
result Khovanov adequacy is explored within the context of independence complexes and homotopy type of extreme spectra.
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 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.
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). 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.
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.
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.
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 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.
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…
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.
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.
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.
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.
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 spectra are shown to be functorial under certain conditions.
problem Understanding functoriality of Khovanov spectra.
method Proving functoriality up to homotopy and sign for Khovanov spectra.
result Khovanov spectra are functorial under specific conditions.
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…
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.
Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate in…
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.
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.
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.
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.
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.