arXiv research

Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.

A new invariant for links generalizes Alexander polynomial for sl_3.

Quantum invariants from $U_h\mathfrak{sl}(2|1)$ are q-holonomic.

Study on quantum $\mathfrak{sl}_3$ invariant for positive links.

Quantum invariants derived from $\mathcal{U}_q(\mathfrak{sl}_2)$ link holonomy.

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd …

Extends knot invariant computation to symmetrically colored sl_N.

This article gives matrix factorizations for the trivalent diagrams and double line appearing in $\mathfrak{sl}_n$ quantum link invariant. These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimens…

We show how to define invariants of graphs related to quantum $\mathfrak{sl}(2)$ when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.

New basis for quantum gl_N invariants derived from Macdonald polynomials.

The abstract discusses new 3-manifold invariants and ETQFTs from Lie superalgebra representations.

New invariants from quantum group theory for hyperbolic 3-manifolds.

Diagrammatic calculus proves Alexander polynomial formulas.

Paper proves zero stability for one-row colored sl₃-Jones polynomials.

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

The paper proves a quantum modularity conjecture for 3-manifolds.

New spin on Khovanov-Rozansky homology categorifies spin link polynomial.

Quantum cluster algebra constructed from web skein relations on surfaces.

Motivated by a possible connection between the $\mathrm{SU}(N)$ instanton knot Floer homology of Kronheimer and Mrowka and $\mathfrak{sl}(N)$ Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yam…

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projecti…

Extends quantum trace map to SL3(C) for 3D surfaces.

Restricts quantum representations of mapping class groups to integral coefficients.

We show that we can release the rigidity of the skew Howe duality process for ${\mathfrak sl}_n$ knot invariants by rescaling the quantum Weyl group action, and recover skein modules for web-tangles. This skew Howe duality phenomenon can be extended to the affine ${\mathfrak sl}_m$ case, corresponding to looking at tan…

Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.

Quantizes Chern-Simons invariant for tangle exteriors.

New invariants derived from Kauffman bracket for 3-manifolds.

We define and study the category of symmetric $\mathfrak{sl}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum $\mathfrak{sl}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric $\mathfrak{sl}_2$-spider is (braided monoidally) equivalent to …

For each finite dimensional, simple, complex Lie algebra $\mathfrak g$ and each root of unity $ξ$ (with some mild restriction on the order) one can define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $τ_M^{\mathfrak g}(ξ)\in \mathbb C$ of oriented 3-manifolds $M$. In the present paper we construct an invariant…

In this paper, we study the quantum $\mathfrak{sl}(n)$ representation category using the web space. Specially, we extend $\mathfrak{sl}(n)$ web space for $n\ge 4$ as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial $P_n(q)$ specialized to a one variable polynomial …

Quantum traces map skein algebras to Fock-Goncharov spaces.

Lectures introduce evaluation of SL(3) foams and link homology.

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$ and a colored link $L$ inside $M$. The link $L$ is split into two parts colored…

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

Modified Hennings invariant defined using quantum groups and integrals.

Geometric model of unbounded sl3 laminations with tropical coordinates.

We define a wall-crossing morphism for Khovanov-Rozansky homology; that is, a map between the KR homology of knots related by a crossing change. Using this map, we extend KR homology to an invariant of singular knots categorifying the Vasilliev derivative of the HOMFLY polynomial, and of $\mathfrak{sl}_n$ quantum invar…

Researchers create a new invariant for knot theory.

We study junctions of Wilson lines in refined SU(N) Chern-Simons theory and their local relations. We focus on junctions of Wilson lines in antisymmetric and symmetric powers of the fundamental representation and propose a set of local relations which realize one-parameter deformations of quantum groups $\dot{U}_{q}(\m…

This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors construct two families of Reshetikhin-Turaev-type invariants of 3-manifolds, $\…

Geometrically describes hyperbolic structures on link complements using quantum groups.

New method calculates Chern-Simons volume for 3-manifolds with surgery diagrams.

Computes Lie algebra structure constants using a graphical calculus.

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the c…

New knot invariants derived using quantum cluster algebras.

The paper clarifies and computes Kashaev-Reshetikhin knot invariants.

Holonomy invariants from $\mathrm{SL}_2(\mathbb{C})$ link complements detect link geometry.

Quantum invariants for fibered links determined by genus and Hopf invariant.

We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We …