arXiv research

In [AU2] we constructed the vacua modular functor based on the sheaf of vacua theory developed in [TUY] and the abelian analog in [AU1]. We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten-Reshetikhin-Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vog…

Study extends knot polynomials to links, identifying them with known invariants.

The multivariable Conway function is generalized to oriented framed trivalent graphs equipped with additional structure (coloring). This is done via refinements of Reshetikhin-Turaev functors based on irreducible representations of quantized gl(1|1) and sl(2). The corresponding face state sum models for the generalized…

Refines a tangle invariant using XC-algebras.

A TQFT is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2+1-cobordism category is built from manifolds which are equipped with an extra structure such as a …

We define a category $v\mathcal{T}$ of tangles diagrams drawn on surfaces with boundaries. On the one hand we show that there is a natural functor from the category of virtual tangles to $v\mathcal{T}$ which induces an equivalence of categories. On the other hand, we show that $v\mathcal{T}$ is universal among ribbon c…

Researchers compare two methods for handlebody constructions, finding they are related with a 'background charge'.

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group $U_qsl_2$ at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin-Turae…

We construct non-semisimple $2+1$-TQFTs yielding mapping class group representations in Lyubashenko's spaces. In order to do this, we first generalize Beliakova, Blanchet and Geer's logarithmic Hennings invariants based on quantum $\mathfrak{sl}_2$ to the setting of finite-dimensional non-degenerate unimodular ribbon H…

Introduces XC-tangles for quantum tangle invariants.

New quantum models unify Alexander and generalized Alexander polynomials for AC links.

We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular categor…

In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of fla…

Internalizes Turaev's construction for TQFTs using ribbon categories.

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…

Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.

We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsváth- Szabó's algebra $\mathcal{B}(n,\mathcal{S})$, an…

Study proves Volume Conjecture for Reshetikhin-Turaev invariants.

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed gr…

Homological blocks match Witten-Reshetikhin-Turaev invariants for Seifert fibered 3-spheres.

We derive an explicit formula for the Witten-Reshetikhin-Turaev SO(3)-invariants of lens spaces. We use the representation of the mapping class group of the torus corresponding to the Witten-Reshetikhin-Turaev SO(3)-TQFT to give such formula.

New knot polynomials reveal patterns and mutations.

Quantum invariants for surfaces in 4D 2-handlebodies.

New algorithm speeds up knot polynomial calculations.

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.

We state a simple criterion to prove the infiniteness of the image of Reshetikhin-Turaev irreducible representations of the mapping class groups of surfaces. We use it to study some of the Reshetikhin-Turaev representations associated to the tori with one and two punctures and derive an alternative proof of a result of…

We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum…

The Witten-Reshetikhin-Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used…

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent.

New model for Witten-Reshetikhin-Turaev invariants using Lagrangian intersections.

We prove that the SU(2) and SO(3) Witten-Reshetikhin-Turaev invariants of any 3-manifold with any colored link inside at any root of unity are algebraic integers.

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 $M\times I$ …

The paper defines a function for knots in Seifert manifolds and connects it to Witten-Reshetikhin-Turaev invariants.

Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the…

Formula calculates MOY webs and link polynomials.

We state Asymptotic Expansion and Growth Rate conjectures for the Witten-Reshetikhin-Turaev invariants of arbitrary framed links in 3-manifolds, and we prove these conjectures for the natural links in mapping tori of finite-order automorphisms of marked surfaces. Our approach is based upon geometric quantisation of the…

Direct proof of Alexander polynomial scaling for L-shaped representations.

Study verifies asymptotic expansion for Reshetikhin-Turaev invariants of fundamental shadow link pairs.

In this paper, we show an isomorphism of homological knot invariants categorifying the Reshetikhin-Turaev invariants for $\mathfrak{sl}_n$. Over the past decade, such invariants have been constructed in a variety of different ways, using matrix factorizations, category $\mathcal{O}$, affine Grassmannians, and diagramma…

We concretely construct a 2-categorically extended TQFT that extends the Reshetikhin-Turaev TQFT to cobordisms with corners. The source category will be a well chosen 2-category of decorated cobordisms with corners and the target bicategory will be the Kapranov-Voevodsky 2-vector spaces.

Manifold calculus of functors, due to M. Weiss, studies contravariant functors from the poset of open subsets of a smooth manifold to topological spaces. We introduce "multivariable" manifold calculus of functors which is a generalization of this theory to functors whose domain is a product of categories of open sets. …

The center Z(C) of a spherical fusion category C (over an arbitrary commutative ring) is modular. We give an algorithm for computing the Reshetikhin-Turaev invariant defined with Z(C). It is based on Hopf diagrams and an explicit description of the structure of the coend of Z(C).

Study on polynomiality and outer nature of functors from Jacobi diagrams to group homomorphisms.

A modular functor is constructed from non-semisimple 3d TFTs.

Paper constructs infinitely many tangent functors on diffeological spaces.

We calculate the homological blocks for Seifert manifolds from the exact expression for the $G=SU(N)$ Witten-Reshetikhin-Turaev invariants of Seifert manifolds obtained by Lawrence, Rozansky, and Mariño. For the $G=SU(2)$ case, it is possible to express them in terms of the false theta functions and their derivatives. …

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this paper we construct various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study …

Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.