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.

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

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

Quantum cluster algebra constructed from web skein relations on surfaces.

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…

Extends knot invariant computation to symmetrically colored sl_N.

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 …

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.

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

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 …

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.

We develop a diagrammatic calculus for representations of unrolled quantum $\mathfrak{sl}_2$ at a fourth root of unity. This allows us to prove Seifert-Torres type formulas for certain splice links using quantum algebraic methods, rather than topological methods. Other applications of this diagrammatic calculus given h…

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

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

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

Geometric model of unbounded sl3 laminations with tropical coordinates.

New basis for quantum gl_N invariants derived from Macdonald polynomials.

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…

Geometrically describes hyperbolic structures on link complements using quantum groups.

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…

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…

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

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

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

Restricts quantum representations of mapping class groups to integral coefficients.

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

The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.

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

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…

We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. We define homological operators and we show that they actually provide a representation for an integral version for $U_q \mathfrak{sl}(2)$. We suggest an isomorp…

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…

Unified approach to deform Lie-Hamilton systems using Poisson-Hopf algebra.

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

Homological model for quantum representations of mapping class groups.

Study of $\mathfrak{sp}_4$-webs on surfaces, proving cluster algebra structure.

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

Quantizes Chern-Simons invariant for tangle exteriors.

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

Using quantum skew-Howe duality, we study the category $\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a ca…

Khovanov homology helps create quantum error-correcting codes.

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…

Study unbounded $\mathfrak{sl}_3$-laminations around punctures.

New invariants derived from Kauffman bracket for 3-manifolds.

The paper calculates colored Jones polynomials for specific link configurations.

Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.

Computes Lie algebra structure constants using a graphical calculus.