Related papers: A refined functorial universal tangle invariant
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…
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…
We define notions of pivotal and ribbon objects in a monoidal category. These constructions give pivotal or ribbon monoidal categories from a monoidal category which is not necessarily with duals. We apply this construction to the braided…
We introduce a class of decorated abstract graphs, that we call XC-tangles, that provides a very convenient framework to study quantum invariants of tangles and virtual tangles. These can be viewed as a far-reaching generalisation of…
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum…
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the…
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…
We prove the unrolled superalgebra $\mathcal{U}_{\xi}^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $\xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its…
The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful monoidal functor of tensor categories, one would like to realize $B$ as category of…
In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the…
Using an extension of the Kontsevich integral to tangles in handlebodies similar to a construction given by Andersen, Mattes and Reshetikhin, we construct a functor $Z:\mathcal{B}\to \widehat{\mathbb{A}}$, where $\mathcal{B}$ is the…
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…
We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under…
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…
The $q$--deformation $U_q (h_4)$ of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra.Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the…
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 $\mathcal{T}$ of handlebody-tangles and present it by generators and…
We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the…
We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We…
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a…
Lyubashenko's construction associates representations of mapping class groups Map_{g,n} of Riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category. We consider this construction as applied to the…