arXiv research

Study reveals $\mathfrak{sl}_2$ action on link homology of specific torus links.

New $\mathfrak{sl}_2$ action on link homologies discovered.

Lifts an $\mathfrak{sl}_2$ action to annular Khovanov homology's stable refinement.

Khovanov-Rozansky homology shows periodic links have group actions.

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

We suggest an index-free formalism allowing to simplify many computations in Riemann geometry. The main ingredients are forms with values in a Clifford algebra and an action of the group $\mathfrak{sl}_2\times \mathfrak{sl}_2$ on such forms.

Quantum cluster algebra constructed from web skein relations on surfaces.

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…

Study HOMFLY-PT homology structure for knots up to 11 crossings.

Foams have Lie algebra symmetries that simplify web state spaces.

Extends Lawrence's representations to integral $U_q \mathfrak{sl}(2)$ Verma-modules and braid groups.

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…

Proofs centerless unimodular contact Lie algebras.

Extends knot invariant computation to symmetrically colored sl_N.

A new invariant for links generalizes Alexander polynomial for sl_3.

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

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.

Geometric model of unbounded sl3 laminations with tropical coordinates.

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

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…

Researchers create a new invariant for knot theory.

We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…

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

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

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…

The paper extends Khovanov homology results to $\mathfrak{sl}(n)$ homologies and provides bounds on knot properties.

We give a purely combinatorial formula for evaluating closed decorated foams. Our evaluation gives an integral polynomial and is directly connected to an integral equivariant version of the $\mathfrak{sl}_N$ link homology categorifying the $\mathfrak{sl}_N$ link polynomial. We also provide connections to the equivarian…

Constructs $y$-ifications of Khovanov homology and proves compatibility with HOMFLY--PT.

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 find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $τ_2\equiv 0$, while the second family has $τ_1\equivτ_2\equiv 0$. The families are differen…

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…

In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we cate…

The paper connects isomonodromic and isospectral deformations for $\mathfrak{sl}_2(\mathbb{C})$ connections.

We define a CR structure on a distinguished hyperplane in $\mathbb{C}^{n+1}$ and the CR sub-Laplacian on this CR manifold. We also define symmetries of the CR sub-Laplacian in general and for this special case construct all of them using the ambient construction. Then we investigate the algebra structure of the symmetr…

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

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 …

Abstract: Bijection strengthened to Morita equivalence integrating Poisson and Cartan-Dirac structures.

Formula for $\mathfrak{sl}_2$ weight system on complete bipartite graphs.

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 …

Homological model for quantum representations of mapping class groups.

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk with marked points. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formula…

New systems of linear PDEs discovered in 3D contact manifolds.

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathf…

Second part of proving linearization theorem for sl2(C).

Study Poisson cohomology and linearize Lie algebra structures.

Study shows link polynomial evaluations from Heegaard Floer theory.