Constructs a 2-functor for tangles extending Burau and Alexander.
problem No specific problem stated; focuses on extending existing concepts.
method Constructs a weak 2-functor from tangles to Lagrangian cospans.
result Extends Burau and Alexander concepts to tangles.
Tangle functors link quantum representations to knot invariants.
problem Constructing tangle functors from semicyclic representations.
method Developed a tangle functor for framed homogeneous tangles colored with semicyclic representations.
result Invariant of (1,1)-tangles from knots matches Kashaev's invariant. Extends sl(3) invariant to tangles via webs and functors.
problem Extending sl(3) invariant to tangles. method Defined a category of sl(3) webs and a functor from tangles to webs. result Recover the sl(3)-polynomial invariant for links from tangles. Refines a tangle invariant using XC-algebras.
problem Building a refined tangle invariant using XC-algebras.
method Constructing a canonical strict monoidal functor that refines the Kerler-Kauffman-Radford invariant.
result Preserves the braiding, twist, and open trace.
New category of virtual tangles connects to existing theories.
problem Characterize virtual tangles and their invariants.
method Define and study a new category vT of virtual tangles, showing it's universal and equivalent to virtual tangles. result Virtual tangles and their invariants extend to framed oriented virtual links.
Introduces XC-tangles for quantum tangle invariants.
problem Quantum invariants of tangles and virtual tangles.
method Introduces XC-tangles and their equivalence to virtual upwards tangles.
result Extension of knot isotopy invariants to XC-tangles.
Extended braid signature to colored tangles, linking to Maslov index and Turaev's functor.
problem Extending braid signature to colored tangles and understanding the homomorphism defect.
method Expressed the defect in terms of the Maslov index and Turaev's Lagrangian functor.
result Defect of multivariable signature expressed using Maslov index and Turaev's functor.
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.
problem Constructing a functor for bottom tangles in handlebodies.
method Extension of Kontsevich integral to handlebodies, functor construction.
result Induces canonical isomorphism between bottom tangles and a specific PROP.
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.
problem Lack of local topological features in evolutionary Khovanov homology for knots and links.
method Introduces a new mathematical framework using persistent Khovanov homology of tangles, employing functor and planar algebra.
result Provides a new method to characterize local features in curve-type data.
Generalizes tangle Floer homology to A-tangles with A-colorings.
problem Extending tangle Floer homology to include A-colorings and cobordisms.
method Defined A-tangles, A-cobordisms, and tangle Floer homology functors for A-modules.
result Constructs tangle Floer homology for A-tangles and shows functoriality.
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.
problem Understanding the structure of knots and links.
method Using quandle categories and tangle decompositions.
result Derived presentations of fundamental quandles for various 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 …
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 M×I …
Study tangle invariants using pillowcase Fukaya category and character varieties.
problem Tangle invariants and their relationships with Khovanov cohomology.
method Define a twisted complex of compact Lagrangians in the pillowcase's Fukaya category, show functoriality, and relate to Khovanov cohomology.
result Hom set with a specific Lagrangian is naturally isomorphic to reduced Khovanov chain complex.
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.
problem Characterizing holonomy-perturbed traceless SU(2) character varieties.
method Examining marked tangles as endomorphisms in the cobordism category and using holonomy-perturbed traceless character variety functor.
result Endomorphisms of immersed curves in the pillowcase have the same image.
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…
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.
Categorifies symmetric functions and computes invariants of tangles.
problem Computing and characterizing invariants of tangles.
method Using homotopy categories and symmetric monoidal categories.
result Existence of symmetric group actions and spectral sequences.
Khovanov homology for pro-tangles and spectral sequences
problem Developing a framework for Khovanov homology for pro-tangles and spectral sequences
method Using pro-tangles, simplicial presheaves, and spectral sequences
result Establishing a fully faithful embedding and an algebraic spectral sequence for pro-tangles
Researchers study rational and pretzel knots using affine group representations.
problem Understanding the structure and properties of rational and pretzel knots.
method Constructing representations of knot groups into the affine group AGL(1,ℂ) via a TQFT valued in spans of singular vector bundles.
result Closed-form expressions for Alexander polynomials and bounds on their zeros for rational and pretzel knots.
Alexander invariant for ribbon tangles connects to planar algebras.
problem Alexander invariant for ribbon tangles.
method Construction of Alexander invariant and its functorial properties.
result Alexander invariant commutes with compositions in a circuit algebra.
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…
New bordered theories for Khovanov homology simplify previous constructions.
problem Simplifying previous constructions of Khovanov homology.
method Formulating Khovanov's invariant in bordered Heegaard Floer homology and providing an alternate construction of Roberts' structures.
result Explicit generators-and-relations description of Hn. The paper develops a new theory for quantum link invariants using quandles and biquandles.
problem Quantum link invariants for links with SL2(C) flat connections. method Using quandles and biquandles, the paper extends Reshetikhin-Turaev functor to tangles.
result A new invariant of links with a gauge class of quandle representations.
A new functor extends Magnus representation to 3D cobordisms.
problem Extending Magnus representation to 3D cobordisms.
method Constructing a monoidal functor from Cob_G to pLagr_R.
result Relates the Magnus functor to the Alexander functor.
Categorifies quantum invariants using cobordism categories and operads.
problem Categorify quantum invariants using cobordism categories and operads.
method Constructs a cobordism category with a colored operad action, categorifies quantum sln invariants. result Consistency of the cobordism category and explicit functor to matrix factorizations conjectured.
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 T of handlebody-tangles and present it by generators and relations. The result tells us that every functor on T 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.
problem Evaluate the homomorphism defect of a signature map for coloured braids.
method Uses a 4-dimensional interpretation of the signature and new 4D tools like the Maslov index and isotropic functor.
result Generalizes the formula of Gambaudo and Ghys to coloured braids and tangles.
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.
problem Constructing TQFTs from ribbon categories.
method Using a special kind of tangle and morphisms defined between tensorial products of the coend, an internal TQFT is constructed.
result An internal TQFT is a lift of Turaev's construction, equivalent to vector spaces when modular.
Enhanced trivalent tangles and handlebody-tangles invariants created.
problem Creating invariants for trivalent and handlebody-tangles.
method Using enhanced trivalent tangles and classical knot theory.
result Constructed invariants for trivalent and handlebody-tangles.
Classifies prime algebraic tangles up to 14 crossings.
problem Classifying prime algebraic tangles systematically.
method Developed a novel canonical representation to distinguish mutant tangles.
result Increased classification of prime tangles up to 14 crossings.
Paper defines new topological invariants for DP tangles.
problem Classifying and understanding doubly periodic tangles.
method Organized components into interlinked compounds; introduced axis-motif.
result Directional type is an invariant of DP tangles.
Simpler method detects trivial rational 3-tangle.
problem Detecting trivial rational 3-tangle.
method Bridge arc replacement method.
result Simpler method detects trivial rational 3-tangle.
New methods found persistent tangles in knots.
problem Persistent tangles in knot diagrams.
method Non-trivial colorings for tangles.
result Any knot with non-trivial coloring has persistent tangles.
Study connects taffy pulling, fractions, and rational tangles.
problem Understanding the relationship between taffy pulling, fractions, and rational tangles.
method Developed a taffy analogue for Conway's characterization of rational tangles and gave a geometric connection.
result Direct geometric connection between rational tangles and taffy pulls.
Method for computing Khovanov homology of tangles.
problem Limited explicit computational studies of Khovanov homology for tangles.
method Arc reduction approach to compute Khovanov homology.
result Derived and computed Poincaré polynomials for simple and complex tangles.
Paper explores conditions for constructing new quasi-alternating links.
problem Applying construction method to non-alternating and opposite type tangles.
method Substituting quasi-alternating crossings with rational tangles of same type.
result Identifies conditions and specific tangles for construction validity.
Invariants for virtual rational tangles defined using Kauffman's bracket.
problem Defining invariants for virtual rational tangles.
method Recursive formula using Kauffman's bracket polynomial.
result Computed several examples of the invariant.