arXiv research

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

The paper calculates a specific weight system for chord diagrams with a particular graph structure.

Constructs a 4-invariant for graphs at c = 3/8.

We introduce a new series $R_k$, $k=2,3,4,\dots$, of integer valued weight systems. The value of the weight system $R_k$ on a chord diagram is a signed number of cycles of even length $2k$ in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small o…

Computes Lie algebra structure constants using a graphical calculus.

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

Quantum cluster algebra constructed from web skein relations on surfaces.

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…

The paper explores weight systems and their applications to graph and embedded graph invariants.

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

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

Develops method to construct Lie algebra weight system kernel using Vogel algebra.

Novel symmetry found in colored HOMFLY polynomials from superalgebras.

Cluster algebras match for specific Lie algebras and surfaces.

Extends knot invariant computation to symmetrically colored sl_N.

A new invariant for links generalizes Alexander polynomial for sl_3.

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

We study the monodromy of meromorphic cyclic $\mathrm{SL}(n,\mathbb{C})$-opers on the Riemann sphere with a single pole. We prove that the monodromy map, sending such an oper to its Stokes data, is an immersion in the case where the order of the pole is a multiple of $n$. To do this, we develop a method based on the wo…

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.

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

Geometric model of unbounded sl3 laminations with tropical coordinates.

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

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

The paper studies geometric structures on SL(n,R) induced by the Killing form.

Researchers create a new invariant for knot theory.

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

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

Extended quantum state result for gl_n weight systems.

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…

Paper defines and computes a new weight system for gl_N Lie algebra.

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…

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

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.

Khovanov-Rozansky homology shows periodic links have group actions.

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 …

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 …

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…

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…

Abstract: Studies systems of equations for pseudo-spherical or spherical surfaces, finding integrability conditions and new families of equations.

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.

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