Researchers prove skein relations for tangle Floer homology.
problem Understanding tangle Floer homology through skein relations.
method Combinatorial proof of skein relations for tangle Floer homology.
result Tangle Floer homology satisfies skein relations.
We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.
Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these b…
We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule define…
Study links using Soergel bimodules and Serre duality.
problem Computing the triply-graded homology of the Whitehead link.
method Representability of partial trace functors and diagrammatics for Hochschild cohomology.
result Computed the triply-graded homology of the Whitehead link.
Study reveals striking uniformity in triply graded link homology for specific braids.
problem Investigating the structure of reduced triply graded link homology in specific degrees.
method Diagrammatic approach to Hochschild cohomology of Soergel bimodules.
result Homology often zero, especially in negative braid case, with striking uniformity.
New invariant for links in handlebodies defined using braid groups and Soergel bimodules.
problem Defining invariants for links in handlebodies.
method Using braid groups and complexes of Soergel bimodules.
result Generalized HOMFLYPT homology for links in handlebodies.
Short note observes quantum Hochschild homology as a composition of known operations.
problem Quantum Hochschild homology as a new invariant of annular links.
method Observes quantum Hochschild homology as a composition of two known operations.
result Quantum Hochschild homology is a valid invariant of annular links.
The paper connects knot homology with sheaf theory and proves symmetry properties.
problem Understanding the Khovanov-Rozansky homology of knots and links.
method Established an isomorphism between link homology and geometric homology, using sheaf theory.
result Proved the qot/q symmetry and computed Khovanov-Rozansky homology of torus links. Defines new Heegaard Floer invariants with actions of both E and F.
problem Developing a new theory of Heegaard Floer invariants with specific actions.
method Introducing larger spaces and bimodule structures with actions of E and F.
result Shows the new bimodules satisfy necessary gluing properties for a 1+1 open-closed TQFT.
We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.
Categorifies symmetric link invariants using foam technology.
problem Categorify symmetric link invariants using slN-webs. method Finite dimensional categorification of symmetric evaluation of slN-webs using foam technology. result Categorifies symmetric link invariants, providing a spectral sequence from Khovanov-Rozansky to symmetric.
Minimal complexes for two-strand braids defined directly.
problem Determining minimal complexes for braids.
method Explicit formulas derived from educated guesswork and reverse engineering.
result Construction of minimal complexes homotopy equivalent to Rickard complexes.
Paper introduces a new algebraic link invariant related to knot Floer and Khovanov homologies.
problem Developing a new algebraic link invariant related to knot homologies.
method Constructing a chain complex C1±1(D) from plat braid diagrams, showing it is isomorphic to Khovanov homology and conjecturing it is a link invariant. result The total homology of the constructed complex is a link invariant, and it is conjectured to be isomorphic to δ-graded knot Floer homology.
The paper proves unique Levi-Civita connections on noncommutative forms.
problem Existence and uniqueness of Levi-Civita connections on noncommutative differential forms.
method Combining Hilbert module and algebraic techniques, proving conditions for existence and uniqueness of Hermitian torsion-free connections.
result Existence and uniqueness of Levi-Civita connections on θ-deformations of compact Riemannian manifolds.
In this paper, we describe a canopolis (i.e. categorified planar algebra) formalism for Khovanov and Rozansky's link homology theory. We show how this allows us to organize simplifications in the matrix factorizations appearing in their theory. In particular, it will put the equivalence of the original definition of Kh…
The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All t…
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with di…
Algorithm computes meridional knot summands in Dehn surgeries.
problem Computing meridional summands in Dehn surgeries for knots.
method Introduced bordered Floer bimodule algorithm for knot complements and meridians.
result Grading of the module computes meridional knot summands in Dehn surgeries.
We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the differe…
New connections found for quantum flag manifolds modules.
problem Unique connections for relative line modules over quantum flag manifolds.
method Applied general results on quantum principal bundles to Heckenberger-Kolb calculi.
result Found bimodule connections with invertible maps.
The paper integrates DGLA to DGLG using HCPs and Hopf algebras.
problem Integrating DGLA to DGLG.
method Definition of DGLG and HCPs, use of graded Hopf algebras.
result Construction of DGLG from DGLA and vice versa.
We characterize the derivation $d:A\to Ω^1_{\der}(A)$ by a universal property introducing a new class of bimodules.
Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…
We show that the triply graded Khovanov-Rozansky homology of the torus link Tn,k stablizes as k→∞. We explicitly compute the stable homology (as a ring), which proves a conjecture of Gorsky-Oblomkov-Rasmussen-Shende. To accomplish this, we construct complexes Pn of Soergel bimodules which categorify t…
Analogous exponential map defined for Hopf algebras.
problem Defining an exponential map for Hopf algebras.
method Analogy with Lie groups, interpretation as states, Hilbert C* bimodules, dual Hopf algebra elements.
result Multiple interpretations of exponential map values in Hopf algebras.
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
Foams 2-equivalent to singular Soergel bimodules.
problem Establishing equivalence between foams and Soergel bimodules.
method 2-equivalence between foams and singular Soergel bimodules of type A.
result 2-category of foams 2-equivalent to singular Soergel bimodules.
New algebras and maps defined in knot Floer homology for trivalent vertices.
problem Categorification of knot Floer homology for trivalent vertices.
method Definition of new algebras, local bimodules, and bimodule maps in bordered knot Floer homology.
result Categorification of representations of U_q(gl(1|1)^-).
This paper develops a theory of graded manifolds in differential geometry.
problem Defining consistent global descriptions of graded manifolds with mixed graded coordinates.
method Using sheaves of graded commutative associative algebras on topological spaces.
result Resolved known issues in the definition of graded manifolds, especially those involving mixed graded coordinates.
Constructs graded jet bundles for Z-graded manifolds and vector bundles.
problem Generalizing jet manifolds to Z-graded structures for differential equations.
method Directly constructs the sheaf of sections of the k-th order jet bundle of a Z-graded vector bundle.
result Establishes a graded version of Atiyah Lie algebroid.
Paper proposes BSP to find stable bimodules of cross-correlated features.
problem Identify groups of features from two data types with strong cross-correlation.
method Iterative-testing based bimodule search procedure (BSP).
result BSP outperforms existing methods in detecting stable bimodules.
This paper proves a conjecture about knot homologies.
problem Proving a spectral sequence from reduced triply graded homology to knot Floer homology.
method Constructing a bigraded spectral sequence from gl0 homology to knot Floer homology. result Proof of the Dunfield-Gukov-Rasmussen conjecture.
We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
Researchers link tangle invariants for Khovanov and knot Floer homologies.
problem Relating tangle invariants for Khovanov and knot Floer homologies.
method Constructing algebraic DA bimodules for tangles and open braids, showing homotopy equivalence to Ozsvath-Szabo bimodules.
result Homotopy equivalence of DA bimodules for tangles and knot Floer homology.
Extends Morse-Novikov Homology to include differential graded coefficients and fibration structures.
problem Extending Morse-Novikov Homology with differential graded coefficients.
method Constructs a Morse-Novikov complex and proves the existence of a Chas-Sullivan-like product for a fibration.
result Proves the existence of a Chas-Sullivan-like product on the Novikov completion of a fibration.
Expands on graded Poisson algebras, their properties, and applications.
problem None explicitly stated; focuses on overview and properties.
method Overview and discussion of properties and applications.
result Provides detailed overview of graded Poisson algebras and their contexts.
The paper solves a problem in constructing a bicategory of algebra bundles.
problem Defining a well-defined composition law for algebra bundles over a smooth manifold.
method Developed a complete solution for a bicategory of algebra bundles, addressing non-invertible bimodules and non-semisimple algebras.
result A complete solution to the problem of constructing a bicategory of algebra bundles.
Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
Constructs a new graded variety from algebraic data.
problem Creating a Z-graded extension of differential varieties. method Algorithm using homotopy retract data of Koszul-Tate resolution.
result Significantly reduced number of homological computations.
Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.
problem Decomposing the Goldman-Turaev Lie bialgebra of a surface.
method Algebraic construction of double Lie bimodules and their combination.
result Decomposes the Goldman-Turaev Lie bialgebra along a simple separating curve.
We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of a N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.
In this work, differential geometry of the Z3-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.
problem Chas-Sullivan products on homology of loop spaces.
method Morse theory with differential graded coefficients, functorial properties, K{ü}nneth formula, Pontryagin-Thom construction.
result Chain-level description of Chas-Sullivan products.
This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
New deformation of link homology for colored diagrams.
problem Understanding colored Khovanov-Rozansky homology.
method Introducing a multi-parameter deformation and extending to braids.
result Link splitting properties and invariants of colored Hopf links.