arXiv research

New Hopf algebras help classify 4D shapes.

Refines a tangle invariant using XC-algebras.

Modified Hennings invariant defined using quantum groups and integrals.

We extend the construction of the Hennings TQFT for ribbon Hopf algebras to the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations proceed in a similar fashion to the ordinary Hopf algebra case, but also require the handling of the non-trivial coassociator in the triple tensor product of the algebr…

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 invariant of framed links is constructed using a ribbon Hopf algebra. In that c…

Knot invariants from XC-structures on Sweedler algebra are trivially determined.

Compute central extension of mapping class group from stated skein algebra

New categories from TQFTs interpret skein relations.

M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that |Z_{Kup}|=|Z_{Henn}|^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.

Constructs TQFTs for cobordisms with cohomology class decorations.

New braided Frobenius algebras created from specific Hopf algebras.

Quantum invariant derived from ternary cohomology of self-distributive structures.

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 class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…

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…

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…

We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…

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…

Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.

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 category of bottom tangles in handlebodies and $\widehat{\mathbb{A}}$ is the degree-com…

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.

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 Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.

We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…

Involutive Hopf monoids yield surface invariants.

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 the additional assumption of factorizability. The resulting functors provide mon…

We prove the unrolled superalgebra $\mathcal{U}_ξ^{H}\mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $ξ$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of…

A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.

New algebraic structure derived from Hopf algebra and Drinfel'd twist.

Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.

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…

Analyses cohomology relations for moving frames and coframes.

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. Th…

Let $n$ be a nonnegative integer, we use ribbon $n-$graph diagrams and the Yamada polynomial skein relations to construct an algebra ${\mathcal Y}_n$ which is shown to be closely related to the Temerley-Lieb Algebra. We prove that the algebra ${\mathcal Y}_2$ is isomorphic to some quotient of a three variables polynomi…

A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T, we prove that the universal invariant J_T of T associated to th…

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomolo…

Invariants for 4-manifolds from Hopf group-algebras.

Analogous exponential map defined for Hopf algebras.

This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.

Paper categorifies a polynomial related to ribbon graphs.

We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator $x$ is invertible and furthermore working polynomials in $\ln x$ instead of polynomials in $x$. We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…

This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) it is shown that up to certain set of local moves, connected simple coverings of B^4 branched over ribbon surfaces, bijectively represent connected orientable 4-dimensional 2-handlebodies up to 2-deforma…

Currents on Lie groups form a Hopf algebra structure.

This paper updates knot invariants using Hopf algebras and categorifies their structure.

This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.

We define an obstruction for a knot to be Z[Z]-homology ribbon, and use this to provide restrictions on the integers that can occur as the triple linking numbers of derivative links of knots that are either homotopy ribbon or doubly slice. Our main application finds new non-doubly slice knots. In particular this gives …