arXiv research

Introduces $q$-transpose for $q$-deformed modular group matrices.

The paper discusses $q$-deformations of the Aomoto complex.

New q-deformed integers help compute Jones polynomials efficiently.

Compactifies stability space for $A_2$ category, introducing $q$-deformed rational numbers.

Link between braid groups and q-deformed rationals solves a classification problem.

Finite specializations of a q-deformed modular group at roots of unity.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.

Unified treatment of gauge theories and Yang-Mills theory duality.

We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.

Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.

Study of $q$-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.

We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double.

The paper constructs TQFTs and Schrödinger representations for Heisenberg group.

We construct a braiding operator in terms of the quantum dilogarithm function based on the quantum cluster algebra. We show that it is a q-deformation of the R-operator for which hyperbolic octrahedron is assigned. Also shown is that, by taking q to be a root of unity, our braiding operator reduces to the Kashaev R-mat…

The theoretical basis for a candidate variational principle for the information bottleneck (IB) method is formulated within the ambit of the generalized nonadditive statistics of Tsallis. Given a nonadditivity parameter $q$, the role of the \textit{additive duality} of nonadditive statistics ($q^*=2-q$) in relating…

Study tangle equations linking enzyme actions to knot theory.

We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also impl…

We developed a strategic of optimal portfolio based on information theory and Tsallis statistics. The growth rate of a stock market is defined by using $q$-deformed functions and we find that the wealth after n days with the optimal portfolio is given by a $q$-exponential function. In this context, the asymptotic optim…

We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $…

We conjecture formulae of the colored superpolynomials for a class of twist knots $K_p$ where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist k…

Holomorphic structures on quantum flag manifolds uniquely defined.

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

The notion of a Kähler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kähler structure has a canonically associated triple satisf…

We compute the noncommutative de Rham cohomology for the finite-dimensional q-deformed coordinate ring $C_q[SL_2]$ at odd roots of unity and with its standard 4-dimensional differential structure. We find that $H^1$ and $H^3$ have three additional modes beyond the generic $q$-case where they are 1-dimensional, while $H…

We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encod…

We extend neural networks with fractional and mixed activation functions for better function approximation.

Algebra Situs is a branch of mathematics which has its roots in Jones' construction of his polynomial invariant of links and Drinfeld's work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum g…

Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.

Constructs explicit nontrivial cycles in Habiro cohomology of smooth varieties.

We study noncommutative bundles and Riemannian geometry at the semiclassical level of first order in a deformation parameter $λ$, using a functorial approach. The data for quantisation of the cotangent bundle is known to be a Poisson structure and Poisson preconnection and we now show that this data defines to a functo…

We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the "refined A…

New complex structures found in quantum SU(3) manifold.

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose-Singer theorem that enlarges the one stated in \cite{Ma2}, and conclude with a differential geometric treatment of KP hierarchy. The example…