arXiv research

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

Geometric model of unbounded sl3 laminations with tropical coordinates.

Extends knot invariant computation to symmetrically colored sl_N.

A new invariant for links generalizes Alexander polynomial for sl_3.

The paper constructs bases for cluster varieties using ${ m SL}_3$-webs and laminations.

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

The paper calculates colored Jones polynomials for specific link configurations.

New model for rational tropical points using $\mathfrak{sp}_4$-webs and measures.

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

Computes Lie algebra structure constants using a graphical calculus.

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

We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and $PGL_m$. The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of laminations. Our construction involves positive configurations of points in the affine bui…

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

Quantum cluster algebra constructed from web skein relations on surfaces.

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

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.

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…

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…

Study surface subgroups acting on projective space, finding bending laminations and spheres.

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

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 …

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 …

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…

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).

Study Poisson cohomology and linearize Lie algebra structures.

Study shows link polynomial evaluations from Heegaard Floer theory.

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

Frobenius homomorphisms for SL_n skein modules generalize knot theory results.

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 …

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.

Extends earthquake and horocycle flows to new measures.

In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators ($\mathcal{D}$-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand pa…

Study of SL(2,R) representations on a once-punctured torus, showing Cantor set spectrum.