A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Lectures introduce evaluation of SL(3) foams and link homology.
Categorifies the skew Howe action on a specific representation category.
We construct the universal sl(2)-tangle cohomology using an approach with webs and dotted foams. This theory depends on two parameters, and for the case of links it is a categorification of the unnormalized Jones polynomial of the link.
In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for N\geq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show…
In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functo…
Study of unoriented SL(4) foams in 3-manifolds.
Categorifies symmetric link invariants using foam technology.
Rewriting theory applied to diagrammatic algebras for categorification.
Spectral sequence connects knot homologies via algebraic geometry.
Quantum method categorifies knot polynomial.
We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Uti…
Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and c…
Quantum groups are realized via surface operator junctions in 4d theory.
Constructs Khovanov homology for links in surfaces, categorifying skein modules.
We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the …
New homology theory for annular knots and links.
This paper contains a categorification of the sl(k) link invariant using parabolic singular blocks of category O. Our approach is intended to be as elementary as possible, providing combinatorial proofs of the main results of Sussan. We first construct an exact functor valued invariant of webs or 'special' trivalent gr…
Defines odd Khovanov homology via categorification of q-Schur algebra.
Estimates the minimum surface area of foam structures.
We give an explicit graded cellular basis of the -web algebra . In order to do this, we identify Kuperberg's basis for the -web space with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…
Paper connects algebraic K-theory to foam geometry.
Generators and relations found for foam categories.
The paper evaluates homology for links in a solid torus with special boundary conditions.
Develops invariants for webs and foams using Seiberg-Witten theory.
Foams have Lie algebra symmetries that simplify web state spaces.
We show how to use Bar-Natan's `divide and conquer' approach to computations to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.
Foams describe singular cobordisms for Khovanov homology.
Study of embedded two-foams from knot diagrams.
We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's.
New homology for links in annulus discovered.
Foam cobordism groups linked to interval exchange automorphisms.
The paper extends foam theory to more complex trivalent graphs.
Formula for evaluating foams related to .
This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex…
Foams 2-equivalent to singular Soergel bimodules.
New 2-representations link spectral enhancements in link homology.
The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted…
The paper constructs semistrict monoidal 2-categories from foam evaluations.
Defines new -foams and their algebras, showing equivalence and applications.
New instanton homology for webs and foams, proving non-vanishing.
We give a purely combinatorial construction of colored link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov-Rozansky homology. Novel f…
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories…
By 2-twist-spinning the knotted graph that represents the knotted handlebody , we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.
We investigate the filtered theory corresponding to the universal sl(2) foam cohomology for links, where a and h are complex numbers. We show that there is a spectral sequence converging to which is invariant under the Reidemeister moves, and whose E1 term is isomorphic to Khovanov homology. This sp…
We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams.
We prove that the foam and matrix factorization universal rational sl3 link homologies are naturally isomorphic as projective functors from the category of link and link cobordisms to the category of bigraded vector spaces.
SU(3) instanton homology counts Tait colorings for webs and foams.