Constructs odd Khovanov homotopy types for links, linking them to even types.
problem Understanding and constructing odd Khovanov homotopy types for links.
method Constructs stable homotopy types X^j_o(L) for links L, with cohomology matching odd Khovanov homology.
result Odd Khovanov homotopy types carry a Z/2 action whose fixed points are related to even Khovanov homotopy types.
Defines homotopy type for links in thickened surfaces.
problem Homotopical Khovanov homology of links in higher genus surfaces.
method Stable homotopy type for links in thickened torus and higher genus surfaces.
result Definition of Khovanov-Lipshark-Sarkar homotopy type for links in thickened surfaces.
Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
New construction of Khovanov homotopy type with stable equivalence results.
problem Constructing and verifying properties of Khovanov homotopy types.
method New construction using Khovanov homotopy type, Burnside category, and products.
result Stable homotopy equivalence of constructed spaces to existing Khovanov homotopy types.
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.
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.
Stability of Khovanov homotopy types of torus links as m increases.
problem Stability of Khovanov homotopy types of torus links as m increases.
method Construction of Khovanov homotopy types and analysis of stabilization.
result Khovanov homotopy types of (n,m) torus links become stably equivalent as mightarrow∞. We give chain homotopy maps of Khovanov-type link homology of a universal differential. The universal differential, discussed by Mikhail Khovanov, Marco Mackaay, Paul Turner and Pedro Vaz, contains the original Khovanov's differential and Lee's differential. We also consider the conditions of any differential ensuring …
New homotopy types defined for links in thickened surfaces with higher genus.
problem Defining stable homotopy types for links in surfaces with higher genus.
method Defined Khovanov-Lipshitz-Sarkar homotopy types and Steenrod squares for links in thickened surfaces with genus > 1.
result First meaningful Khovanov-Lipshitz-Sarkar stable homotopy types for links in 3-manifolds other than the 3-sphere.
Notes on Khovanov and knot Floer theories' stable homotopy types.
problem Understanding stable homotopy types in Khovanov and knot Floer theories.
method Introduction to Khovanov and knot Floer theories' stable homotopy types.
result Introduction of stable homotopy types in Khovanov and knot Floer theories.
New invariant for 4-manifolds with framed links, stronger than existing invariants.
problem Distinguishing 4-manifolds with framed links.
method Introducing a new invariant called KLS lasagna homotopy type.
result The new invariant is stronger than existing invariants.
Extends Khovanov cohomology to colored links using stable homotopy types.
problem Extending Khovanov cohomology to colored links.
method Defining a stable homotopy type X_col(L_c) for colored links L, using categorified Jones-Wenzl projectors and infinite torus braids.
result Computes stable homotopy types for specific colored links and makes a conjecture for others.
The paper defines a new homotopy type for colored links and proves stabilization behavior.
problem Understanding the behavior of Khovanov homotopy types for colored links.
method Definition of a Khovanov homotopy type for colored links and quantum spin networks, and derivation of its properties.
result Stabilization of the homotopy types for n-colored B-adequate links as nightarrow∞. Study homotopy type of periodic links using Khovanov spectrum.
problem Understanding the homotopy type of periodic links.
method Construct Khovanov spectrum and analyze its cohomology.
result Khovanov spectrum admits a homology group action and Steenrod algebra structure.
New stable homotopy types for knots and diagrams, aiding in their computation.
problem Computing Khovanov-Rozansky cohomology for knots and diagrams.
method Lifted simplified Khovanov-Rozansky complexes to stable homotopy types for open 2-braids.
result New stable homotopy types for knots and diagrams exhibit interesting behavior.
New algorithm calculates Steenrod squares in Khovanov cohomology.
problem Computing Steenrod squares in Khovanov cohomology.
method Flow category simplification techniques to calculate second Steenrod square and Bockstein homomorphisms.
result Observation of new homotopy types and evidence against CP2 summands. 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.
Inspired by bordered Floer homology, we describe a type A structure on a Khovanov homology for a tangle, which complements the type D structure in a previous paper. The type A structure is a differential module over a certain algebra. This can be paired with the type D structure to recover the Khovanov chain complex. T…
Paper conjectures and proves simplicial complexes are homotopy equivalent to spheres, linking to knot theory.
problem Homotopy type of circle graphs complexes.
method Constructing and analyzing simplicial complexes from circle graphs.
result Simplicial complexes are homotopy equivalent to spheres in many cases.
Given a link diagram L we construct spectra X^j(L) so that the Khovanov homology Kh^{i,j}(L) is isomorphic to the (reduced) singular cohomology H^i(X^j(L)). The construction of X^j(L) is combinatorial and explicit. We prove that the homotopy type of X^j(L) depends only on the isotopy class of the corresponding link.
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…
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.
Proves Serre duality in Khovanov-Rozansky homology.
problem Relating top and bottom Hochschild degrees in Khovanov-Rozansky homology.
method Using type A Soergel bimodules and full twist as a Serre functor.
result Categorifies Kálmán's theorem on Hochschild degrees.
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.
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…
Refines Khovanov homology using stable homotopy theory.
problem Khovanov homology needs refinement.
method Stable homotopy refinement approach.
result Stable homotopy refinement of Khovanov homology constructed.
New invariant constructed using stable homotopy methods.
problem Constructing a new link invariant.
method Stable homotopy theory and Khovanov's method.
result A Khovanov slk-stable homotopy type constructed. Analogous Morse moves for flow categories simplify flow data.
problem Simplifying flow data in Morse theory.
method Developed moves for framed flow categories analogous to Whitney trick and handle cancellation.
result Simplified flow categories can be computed by hand.
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.
We describe a bordered version of totally twisted Khovanov homology. We first twist Roberts's type D structure by adding a "vertical" type D structure which generalizes the vertical map in twisted tangle homology. One of the distinct advantages of our type D structure is that it is homotopy equivalent to a type $…
New homotopy refinements for tangle invariants defined.
problem Stable homotopy refinements for tangle invariants.
method Defined stable homotopy refinements of Khovanov's arc algebras and tangle invariants.
result Stable homotopy refinements of Khovanov's arc algebras and tangle invariants defined.
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.
Paper introduces moves to simplify framed flow categories.
problem Simplifying framed flow categories for easier study.
method Inspired by Morse-Smale moves, introduces moves to change framed flow categories without altering their stable homotopy type.
result Finite sequence of moves can connect two framed flow categories representing the same stable homotopy type.
Infinite braids' categorification proven using Khovanov homology.
problem Categorifying Jones-Wenzl projectors for infinite braids.
method Limiting Khovanov chain complex and homotopy types.
result Jones-Wenzl projectors categorified for infinite braids.
We describe the first part of a gluing theory for the bigraded Khovanov homology with integer coefficients. This part associates a type D structure to a tangle properly embedded in a half-space and proves that the homotopy class of the type D structure is an invariant of the isotopy class of the tangle. The constructio…
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.
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.
Geometrically realizes Khovanov homology for semiadequate links.
problem Computing Khovanov homology for semiadequate links.
method Introducing partial presimplicial sets and their geometric realization.
result Concrete formula for homotopy type of geometric realization.
In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, …
New proof of Khovanov spectrum equivalence at extreme grading.
problem Proving homotopy equivalence of spectra at extreme quantum grading.
method Stable homotopy equivalence proof using González-Meneses et al. spectrum and Lipshitz-Sarkar Khovanov spectrum.
result Stable homotopy equivalence between the two spectra at extreme quantum grading.
We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.
Circle graph complexes reveal link properties via Khovanov homology.
problem Understanding topological properties of links using circle graph complexes.
method Analyzing homotopy types and computing Khovanov homology of specific knots.
result Real extreme Khovanov homology of a 4-strand pretzel knot computed.
New homotopy refinements for tangle invariants.
problem Stable homotopy refinements for tangle invariants.
method Refined Khovanov and Chen-Khovanov spectra.
result Induces refinements of platform algebras and invariants.
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.
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.
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.
Paper proves Khovanov homology can be a TQFT.
problem Proving Khovanov homology can be a TQFT.
method Modified Khovanov homology for Frobenius algebras of rank 2, producing a monoidal functor.
result Khovanov homology can be a TQFT.