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.
Constructs a 2-functor for tangles extending Burau and Alexander.
Tangle functors link quantum representations to knot invariants.
Extends invariant to tangles via webs and functors.
Refines a tangle invariant using XC-algebras.
New category of virtual tangles connects to existing theories.
Introduces XC-tangles for quantum tangle invariants.
We construct a functor from the category of oriented tangles in R^3 to the category of Hermitian modules and Lagrangian relations over Z[t,t^{-1}]. This functor extends the Burau representations of the braid groups and its generalization to string links due to Le Dimet.
A new integral for tangles in handlebodies connects to Hopf algebras.
A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of…
Develops persistent Khovanov homology for tangles.
Generalizes tangle Floer homology to A-tangles with A-colorings.
In this article we extend evaluations of the Kauffman bracket on regular isotopy classes of knots and links to a variety of functors defined on the category of framed tangles. We show that many such functors exist, and that they correspond up to equivalence to bilinear forms on free, finitely-generated modules over com…
Fundamental quandle helps in understanding knots and links.
A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangl…
The involutory birack counting invariant is an integer-valued invariant of unoriented tangles defined by counting homomorphisms from the fundamental involutory birack of the tangle to a finite involutory birack over a set of framings modulo the birack rank of the labeling birack. In this first of an anticipated series …
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting propertie…
There are two categorifications of the Jones polynomial: "even" discovered by M.Khovanov in 1999 and "odd" dicovered by P.Ozsvath, J.Rasmussen and Z.Szabo in 2007. The first one can be fully constructed in the category of cobordisms (strictly: in the additive closure of that category), where we can build a complex for …
Study tangle invariants using pillowcase Fukaya category and character varieties.
The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface …
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. A…
Study character varieties of tangles to map immersed curves in the pillowcase.
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological cobordisms and show that it is 2-commutative: the composition of 2-morphisms al…
Taking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considerin…
New quantum models unify Alexander and generalized Alexander polynomials for AC links.
Categorifies symmetric functions and computes invariants of tangles.
Khovanov homology for pro-tangles and spectral sequences
Researchers study rational and pretzel knots using affine group representations.
Alexander invariant for ribbon tangles connects to planar algebras.
The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an e…
The paper develops a new theory for quantum link invariants using quandles and biquandles.
A new functor extends Magnus representation to 3D cobordisms.
Categorifies quantum invariants using cobordism categories and operads.
We describe how to formulate Khovanov's functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts' Type D and Type A structures in Khovanov homology, and his algebra , in terms of Khovanov's theory of modules over …
A handlebody-knot is a handlebody embedded in the 3-sphere. We establish a uniform method to construct invariants for handlebody-links. We introduce the category of handlebody-tangles and present it by generators and relations. The result tells us that every functor on that gives rise to inv…
We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a c…
Extends a formula for the homomorphism defect of a signature map to coloured braids.
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…
Internalizes Turaev's construction for TQFTs using ribbon categories.
Enhanced trivalent tangles and handlebody-tangles invariants created.
Classifies prime algebraic tangles up to 14 crossings.
Paper defines new topological invariants for DP tangles.
Simpler method detects trivial rational 3-tangle.
New methods found persistent tangles in knots.
Study connects taffy pulling, fractions, and rational tangles.
Method for computing Khovanov homology of tangles.
Invariants for virtual rational tangles defined using Kauffman's bracket.
Paper explores conditions for constructing new quasi-alternating links.