Defines new g l 2 \mathfrak{gl}_2 gl 2 -foams and their algebras, showing equivalence and applications.
problem Developing new algebraic structures for g l 2 \mathfrak{gl}_2 gl 2 . method Introducing parameter-dependent g l 2 \mathfrak{gl}_2 gl 2 -foams and their associated web and arc algebras. result All specializations of these algebras are equivalent, with applications in link and tangle invariants.
New homology for links in annulus discovered.
problem Equivariant g l N \mathfrak{gl}_N gl N homology for links in annulus. method Foam evaluation for equivariant g l N \mathfrak{gl}_N gl N homology. result Introduced new homology for annular links.
Foams have Lie algebra symmetries that simplify web state spaces.
problem Understanding symmetries in foam structures.
method Defined an action of a Lie subalgebra on foams compatible with g l N \mathfrak{gl}_N gl N -foam evaluation. result Endows g l N \mathfrak{gl}_N gl N -web state spaces with s l 2 \mathfrak{sl}_2 sl 2 -action. Using quantum skew-Howe duality, we study the category Rep ( g l ( m ∣ n ) ) \operatorname{Rep}(\mathfrak{gl}(m|n)) Rep ( gl ( m ∣ n )) of tensor products of exterior powers of the standard representation of U q ( g l ( m ∣ n ) ) U_q(\mathfrak{gl}(m|n)) U q ( gl ( m ∣ n )) , and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a ca…
Odd Khovanov homology gets a new algebraic action from super foams.
problem Understanding the algebraic structure of odd Khovanov homology.
method Introducing a local g l 1 ∣ 1 \mathfrak{gl}_{1|1} gl 1∣1 -action on odd Khovanov homology via super foams. result The action of g l 1 ∣ 1 \mathfrak{gl}_{1|1} gl 1∣1 on odd Khovanov homology is shown to arise from super foams. Develops higher representation theory for odd Khovanov homology and rewriting theory.
problem Quantum topology and higher algebraic structures.
method Higher representation theory and rewriting theory applied to Khovanov homology.
result Established a basis theorem for graded g l 2 \mathfrak{gl}_2 gl 2 -foams. Witt algebra acts on categorified quantum groups in type A.
problem Action of Witt algebra on categorified quantum groups.
method Construction of action on categorified quantum group and foams.
result Action of Witt algebra on foams recovers previous results.
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 g l 0 \mathfrak{gl}_0 gl 0 homology to knot Floer homology. result Proof of the Dunfield-Gukov-Rasmussen conjecture.
Formula for evaluating foams related to s l N \mathfrak{sl}_N sl N .
problem Evaluating foams decorated with s l N \mathfrak{sl}_N sl N . method Purely combinatorial formula connecting to link homology.
result Integral polynomial evaluation directly related to link homology.
Rewriting theory applied to diagrammatic algebras for categorification.
problem Finding bases in graded g l 2 \mathfrak{gl}_2 gl 2 -foams. method Algorithmic approach combining linear and higher rewriting, modulo rules capturing categorical properties.
result First proof of a basis theorem for graded g l 2 \mathfrak{gl}_2 gl 2 -foams. We give a purely combinatorial construction of colored s l n \mathfrak{sl}_n sl n link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between s l n \mathfrak{sl}_n sl n webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel f…
Defines odd Khovanov homology via categorification of q-Schur algebra.
problem Constructing odd Khovanov homology via representation theory.
method Supercategorification of q-Schur algebra, odd foams, tensor product on chain complexes.
result Odd Khovanov homology defined via categorification.
Categorifies symmetric link invariants using foam technology.
problem Categorify symmetric link invariants using s l N \mathfrak{sl}_N sl N -webs. method Finite dimensional categorification of symmetric evaluation of s l N \mathfrak{sl}_N sl N -webs using foam technology. result Categorifies symmetric link invariants, providing a spectral sequence from Khovanov-Rozansky to symmetric.
Researchers compute g l 2 \mathfrak{gl}_2 gl 2 -skein modules for lens spaces.
problem Computing g l 2 \mathfrak{gl}_2 gl 2 -skein modules for lens spaces. method Action of g l 2 \mathfrak{gl}_2 gl 2 -skein algebra on solid torus's g l 2 \mathfrak{gl}_2 gl 2 -skein module. result Lens spaces' g l 2 \mathfrak{gl}_2 gl 2 -skein modules span by specific elements. New basis for quantum gl_N invariants derived from Macdonald polynomials.
problem Constructing new bases for quantum gl_N invariants.
method Using interpolation Macdonald polynomials and Okounkov's results.
result Cyclotomic expansions for gl_N invariants and knot invariants.
In this paper we define an explicit basis for the g l n \mathfrak{gl}_n gl n -web algebra H n ( k ⃗ ) H_n(\vec{k}) H n ( k ) (the g l n \mathfrak{gl}_n gl n generalization of Khovanov's arc algebra) using categorified q q q -skew Howe duality. Our construction is a g l n \mathfrak{gl}_n gl n -web version of Hu--Mathas' graded cellular basis and has two major application…
Lectures introduce evaluation of SL(3) foams and link homology.
problem Categorification of quantum s l 3 \mathfrak{sl}_3 sl 3 web and link invariant. method Introduction and review of S L ( 3 ) \mathsf{SL}(3) SL ( 3 ) foams and their evaluation. result Categorification of Kuperberg quantum s l 3 \mathfrak{sl}_3 sl 3 web and link invariant. New structure for quantum algebra representations.
problem Understanding representations of quantum algebras.
method Constructing a quotient category of annular quantum g l n \mathfrak{gl}_{n} gl n webs. result Equivalent to finite dimensional representations of quantum Levi subalgebras.
The paper constructs semistrict monoidal 2-categories from foam evaluations.
problem Creating examples of semistrict monoidal 2-categories.
method Using a closed foam evaluation formula as input, the paper rigorously constructs semistrict monoidal 2-categories.
result The constructed monoidal 2-categories are semistrict, have duals and adjoints, and carry a spatial duality structure.
Simplified computation of symmetric gl_1 homology for links.
problem Computing the symmetric gl_1 homology for uncolored links.
method Down-to-earth description, basis construction, and algorithm for computation.
result An algorithm and program for computing the invariant for uncolored links.
Spectral sequence connects knot homologies via algebraic geometry.
problem Connecting algebraic and geometric knot homologies.
method Bigraded spectral sequence from gl(0)-homology to knot Floer homology.
result Constructs a Bockstein-type spectral sequence.
Paper defines and computes a new weight system for gl_N Lie algebra.
problem Understanding the weight system of Lie algebra gl_N.
method Two approaches: Kazarian's invariant and Harish-Chandra isomorphism.
result Computes the gl_N weight system on chord diagrams.
We use the technique of quantum skew Howe duality to investigate the monoidal category of exterior powers of the standard representation of U q ( g l ( 1 ∣ 1 ) ) U_q(\mathfrak{gl}(1|1)) U q ( gl ( 1∣1 )) . This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family o…
In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for s l n \mathfrak{sl}_n sl n . Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category O \mathcal{O} O , affine Grassmannians, and diagramma…
Extended quantum state result for gl_n weight systems.
problem Quantum states associated with gl_n weight systems.
method Extended Corfield et al. result to all gl_n weight systems.
result All gl_n weight systems are quantum states.
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
A moduli space is constructed for affine homogeneous spaces using their local geometry.
problem Describing and classifying affine homogeneous spaces locally and globally.
method Local description via curvature, torsion, and connection; algebraic variety construction; Spencer cohomology for infinitesimal deformations.
result A moduli space M ( g l V ) \mathfrak{M}(\mathfrak{gl}\,V) M ( gl V ) is constructed for affine homogeneous spaces of dimension d i m V dim V d imV . Constructs Khovanov homology for links in surfaces, categorifying skein modules.
problem Categorify skein modules for surfaces.
method Functorial Khovanov homology, gl(2) foams, tensor product, bifunctor.
result Candidate bifunctor extends to a monoidal structure, verifying positivity conjectures.
State sums for quantum link invariants from a specific representation.
problem Calculating quantum link invariants from a specific representation of U_q(gl_{N|M}).
method Using state sums and representation theory of U_q(gl_{N|M}).
result Explicit relation with Kashaev invariants for the N-th exterior power of the standard representation.
New geometric method constructs singular Gelfand-Tsetlin modules.
problem Constructing 1-singular Gelfand-Tsetlin modules.
method Using complex geometry and universal ring D o \mathcal D_o D o with vector space S \mathcal S S . result Obtained a construction of the universal 1-singular Gelfand-Tsetlin g l n ( C ) \mathfrak{gl}_n(\mathbb C) gl n ( C ) -module. Authors compute stable homology of torus knots using a new deformation technique.
problem Computing stable homology of torus knots.
method Link-splitting deformation (y-ification) of link homology.
result Explicit computation of y y y -ified g l N \mathfrak{gl}_N gl N stable Khovanov--Rozansky homology of torus knots. Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).
problem Understanding the Burau representation and its relation to knot Floer homology.
method Developed a Heegaard Floer homology theory and associated a bordered sutured Heegaard Floer homology group to any tangle.
result Established a connection between the Burau representation and quantum gl(1|1), leading to a geometric proof of the braid representation.
The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, introduced by Yu.I.Manin. We prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak {gl}_{m…
New 2-representations link spectral enhancements in link homology.
problem Spectral enhancements in link homology.
method Skew Howe duality, frames, and multifunctors.
result Spectral 2-representations of categorified quantum groups.
New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.
problem Constructing a 3-manifold invariant from Viro's gl(1|1) polynomial.
method Following Costantino, Geer, and Patureau-Mirand's method in relative G-modular categories.
result The invariant can distinguish homotopy equivalent lens spaces.
Defines Khovanov homology with gl(1|1) action, preserving Reidemeister moves.
problem Khovanov homology with gl(1|1) action definition and preservation.
method Definition of annular odd Khovanov homology, proof of gl(1|1) action preservation.
result Annular odd Khovanov homology carries an action of gl(1|1) preserved by Reidemeister moves.
We use super q q q -Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of g l N \mathfrak{gl}_N gl N -modules (and, more generally, g l N ∣ M \mathfrak{gl}_{N|M} gl N ∣ M -modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an ap…
The paper connects knot Floer homology to quantum representations.
problem Decategorification of knot Floer homology using bordered theory.
method Relating decategorifications to representations of U_q(gl(1|1)).
result Identifies decategorifications with tensor products of representations.
We give an explicit graded cellular basis of the s l 3 \mathfrak{sl}_3 sl 3 -web algebra K S K_S K S . In order to do this, we identify Kuperberg's basis for the s l 3 \mathfrak{sl}_3 sl 3 -web space W S W_S W S with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…
New quantum integrals discovered for a spin chain model.
problem Exploring quantum integrals for a spin chain model.
method Using surface defects and observables in 4D N = 2 {\mathcal{N}}=2 N = 2 super-QCD. result First construction of quantum integrals and their joint eigenvectors.
New s l 2 \mathfrak{sl}_2 sl 2 action on link homologies discovered.
problem Understanding symmetries in link homologies.
method Constructing s l 2 \mathfrak{sl}_2 sl 2 action on equivariant g l N \mathfrak{gl}_N gl N -link homologies. result Obtained s l 2 \mathfrak{sl}_2 sl 2 action and p p p -DG structures. In this paper we use Kuperberg's s l 3 \mathfrak{sl}_3 sl 3 -webs and Khovanov's s l 3 \mathfrak{sl}_3 sl 3 -foams to define a new algebra K S K^S K S , which we call the s l 3 \mathfrak{sl}_3 sl 3 -web algebra. It is the s l 3 \mathfrak{sl}_3 sl 3 analogue of Khovanov's arc algebra. We prove that K S K^S K S is a graded symmetric Frobenius algebra. Furthermore, we cate…
New bounds on virtual link genus using quantum supergroups.
problem Finding strong lower bounds on the minimal genus of virtual links.
method Defined a U q ( g l ( m ∣ n ) ) U_q(\mathfrak{gl}(m|n)) U q ( gl ( m ∣ n )) invariant equivalent to the CSW polynomial, generalized to all U q ( g l ( m ∣ n ) ) U_q(\mathfrak{gl}(m|n)) U q ( gl ( m ∣ n )) . result Generalized CSW lower bounds to all quantum supergroups U q ( g l ( m ∣ n ) ) U_q(\mathfrak{gl}(m|n)) U q ( gl ( m ∣ n )) with m , n > 0 m,n>0 m , n > 0 . Study quantized Coulomb branches of Jordan quiver gauge theories and their connections to Cherednik algebras.
problem Understanding the structure of quantized Coulomb branches in Jordan quiver gauge theories.
method Proved isomorphisms between quantized Coulomb branches and spherical graded Cherednik and cyclotomic rational Cherednik algebras.
result Quantized Coulomb branches are deformations of subquotients of Yangians of affine g l ( 1 ) \mathfrak{gl}(1) gl ( 1 ) . New quantum models unify Alexander and generalized Alexander polynomials for AC links.
problem Defining and distinguishing AC links and virtual knots.
method Generalizing AC links to virtual tangles and using quantum supergroups.
result Generalized Alexander polynomials are distinct from Alexander polynomials for AC links.
Study of surface defects in gauge theories leads to duality and separation of variables.
problem Understanding surface observables and their transitions in gauge theories.
method Utilized Fourier transformations and spectral problems to derive dualities and separation of variables.
result Exact duality between spectral problems of spin chains and Gaudin models.
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.
Unified construction of link homologies using categorical annular evaluation.
problem Unified construction of link homologies.
method Categorical annular evaluation.
result Unified construction of s l n \mathfrak{sl}_n sl n and HOMFLYPT Khovanov-Rozansky link homologies.