Related papers: A graphical categorification of the two-variable C…

We construct a representation of the braid groups in a cluster C*-algebra coming from a triangulation of the Riemann surface S with one or two cusps. It is shown that the Laurent polynomials attached to the K-theory of such an algebra are…

We show that a projective space P^\infty(Z/2) endowed with the Alexandrov topology is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over…

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron…

Based on the q-difference operators for the genus-two skein algebra, we show the correspondence between the reduced Askey-Wilson polynomials and the skein module of the genus-two handlebody.

We begin by considering a sequence of polynomials in three variables whose coefficients count restricted binary overpartitions with certain properties. We then concentrate on two specific subsequences that are closely related to the…

Given a reciprocal/palindromic polynomial of even degree, we show that the gamma vector is essentially given by an inverted Chebyshev polynomial basis expansion. As an immediate consequence, we characterize real-rootedness of a linear…

A skew Laurent polynomial ring R[x^{\pm 1};\alpha] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for…

We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type. Each of these cluster algebras which we call tubular…

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [arXiv:2101.00282, arXiv:2107.10106]. Here we define new web models corresponding to each of the rank-two…

We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we…

Let $X$ be a projective variety with an isolated $A_2$ singularity. We study its bounded derived category and prove that there exists a crepant categorical resolution $\pi_*\colon \widetilde{\mathcal{D}} \to D^b(X)$, which is a Verdier…

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several…

We study quartic double fivefolds from the perspective of Fano manifolds of Calabi-Yau type and that of exceptional quaternionic representations. We first prove that the generic quartic double fivefold can be represented, in a finite number…

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial…

We define a class of multivariate Laurent polynomials closely related to Chebyshev polynomials, and prove the simple but somewhat surprising (in view of the fact that the signs of the coefficients of the Chebyshev polynomials themselves…

We show that the associativity condition of the universal symmetric 2-algebraic 2-valued group defined by the Buchstaber polynomial admits several mutually equivalent interpretations from the viewpoints of the Chazy equation, Gauss-Manin…

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…

We prove a formula expressing the Kerov polynomial $\Sigma_k$ as a weighted sum over the lattice of noncrossing partitions of the set $\{1,...,k+1\}$. In particular, such a formula is related to a partial order $\mirr$ on the Lehner's…

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by…