arXiv research

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

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

New q-deformed integers help compute Jones polynomials efficiently.

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

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

Study tangle equations linking enzyme actions to knot theory.

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

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

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

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

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.

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…

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

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…

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…

The study calculates the average genus of rational knots and links.

New method for simplifying knots with specific properties.

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 describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…

New findings on prime theta-curves with simple tangles.

The paper calculates the number of oriented rational links with a given deficiency.

New knot invariant λ bounds rational unknotting.

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…

A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding…

Classifies surgeries on torus knots and cables that bound rational homology balls.

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rat…

Classifies worst approximable rational numbers using hyperbolic geometry.

The paper calculates bounds for unknotting rational tangles using knot Floer homology.

This paper proves an upper limit on rational points on curves.

Formula conjectured for rational cuspidal curves in projective plane.

New examples show deletion type admissible pairs can be rigid under rational saturation.

New method uses rational Witt span to bound concordance crosscap number of knots.

Classifies fertility of all rational links.

Introducing a way to modify knots using $n$-trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and Yamada). The same result is shown for 4-genera and finite reductions of the homolog…

Paper finds linking numbers for Montesinos links using a simple algorithm.

The article calculates a multiplying factor to convert rational Vassiliev invariants to integer-valued ones.

In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the …

We note that a rational $3$-tangle diagram is obtained from a combination of four generators. There is an algorithm to distinguish two rational $3$-tangle diagrams up to isotopy. However, there is no perfect classification about rational $3$-tangle diagrams such as the classification of rational $2$-tangle diagrams cor…

There is a natural way to associate with a transformation of an isotopy class of rational tangles to another, an element of the modular group. The correspondence between the isotopy classes of rational tangles and rational numbers follows, as well as the relation with the braid group $B_3$.

In this paper, we introduce a rational $τ$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsváth-Szabó contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives o…

We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, a knot) in terms of a special continued fraction for the rational number that defines the given link.

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…

New geometric proof for rational tangles links-quivers correspondence.

Study shows no hyperkähler fourfolds in specified conditions.