Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
New q-deformed integers help compute Jones polynomials efficiently.
problem Computing Jones polynomials of rational links efficiently.
method Defining q-deformed integers from pairs of coprime integers and using them to compute Jones polynomials.
result Efficient algorithm for computing Jones polynomials of rational links.
Compactifies stability space for A2 category, introducing q-deformed rational numbers.
problem Stability conditions in triangulated categories and their compactifications.
method Embedding into an infinite-dimensional projective space, using B3 braid group action. result Two orbits in the boundary correspond to q-deformed rational numbers. Link between braid groups and q-deformed rationals solves a classification problem.
problem Classifying faithful complex specializations of the Burau representation of braid group B3.
method Established a link between Burau representation and q-deformed rational numbers.
result Proved faithfulness of Burau representation outside a specific annulus.
Study of q-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.
problem Geometry of q-rationals and their properties. method Construction and analysis of deformed Farey triangulation and deformed modular surface; definition and study of Springborn operations.
result Derivation of a formula for the q-deformed midpoint and new q-deformation of Markov numbers. Study tangle equations linking enzyme actions to knot theory.
problem Proving the Jones Unknot conjecture and understanding tangle solutions.
method Analyzing framed tangle equations and introducing Kauffman bracket ratios.
result Unique rational solutions for tangle equations imply the Jones Unknot conjecture.
Finite specializations of a q-deformed modular group at roots of unity.
problem Understanding the finiteness of specializations of a q-deformed modular group at roots of unity.
method Introduced a q-deformed modular group and studied its specializations at roots of unity.
result For ζn being a primitive nth root of unity, PSLq(2,Z)∣q=ζn is finite if and only if Gq(ζn) is finite. Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
The paper discusses q-deformations of the Aomoto complex.
problem Deformation of cochain complexes associated with hyperplane arrangements.
method Replaces entries of coboundary maps with q-analogues and analyzes the resulting structures. result The q-deformation can be a cochain complex under certain conditions and yields local system cohomology groups. Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.
problem Existence and uniqueness of solutions to loop equations in generalized Frobenius manifolds.
method Proves existence and uniqueness of solutions to loop equations for semisimple generalized Frobenius manifolds.
result Existence and uniqueness of solutions to loop equations for semisimple generalized 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.
Geometrically, Legendrian surfaces related by surgery have related skein-valued cluster spaces.
problem Understanding the skein-valued cluster transformation in Legendrian surfaces.
method Geometric considerations of moduli of holomorphic curves.
result Skein-valued cluster transformation of Legendrian surfaces related by surgery.
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.
problem Constructing TQFTs and Schrödinger representations for Heisenberg group.
method Using Lagrangian correspondences and q-deformation of U(1).
result Normalization of Schrödinger bimodule action reproduces abelian TQFT.
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…
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 A2 clasps in the Karoubi envelope of A2 spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type A2. The A2 spider is a diagrammatic description of the representation category for Uq(sl3) and the $…
We conjecture formulae of the colored superpolynomials for a class of twist knots Kp 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.
problem Defining unique holomorphic structures on quantum flag manifolds.
method Constructing covariant q-deformed holomorphic structures. result Holomorphic structures are unique for simple relative Hopf modules.
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
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…
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
We compute the noncommutative de Rham cohomology for the finite-dimensional q-deformed coordinate ring Cq[SL2] at odd roots of unity and with its standard 4-dimensional differential structure. We find that H1 and H3 have three additional modes beyond the generic q-case where they are 1-dimensional, while $H…
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
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.
problem Limitations in approximating higher-order smooth functions in complex spaces.
method Incorporating fractional exponents in activation functions and defining new density functions.
result Improved accuracy and broader applicability of neural network approximation theory.
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…
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature (3,2). method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤5 in the quadric. New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.
The study calculates the average genus of rational knots and links.
problem Finding the average genus of rational knots and links.
method Enumerating and calculating the number of rational knots and links with a given crossing number.
result A precise formula for the average minimal genus of rational knots and links.
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…
New rational band moves simplify knot classification.
problem Classifying knots using rational moves.
method Introduced oriented rational band moves and proved their effectiveness.
result Knots that can be unlinkified by rational moves are rationally slice.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
Jones polynomial coincidences explored for rational knots.
problem Identifying coincidences in Jones polynomial of rational knots.
method Moves on continued fraction expansion of rational knots, conjectured to generate all coincidences.
result Conjectured moves are sufficient to generate all Jones rational coincidences.
Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
problem Integrable hierarchies for generalized Frobenius manifolds with non-flat unity.
method Constructs a bihamiltonian integrable hierarchy of hydrodynamic type.
result Integrable hierarchy possesses Virasoro symmetries and a tau structure.
Lower bounds on rational slice genus using Heegaard Floer invariants.
problem Measuring complexity of homology classes in 4-manifolds.
method Introducing rational slice genus, bounding Heegaard Floer τ invariants, using satellite links and closed braids.
result Lower bounds on rational slice genus in terms of Heegaard Floer τ invariants.
This paper gives two new combinatorial topological proofs of the classification of rational tangles. Each proof rests on an elegant lemma showing that rational tangles are isotopic to canonical alternating rational tangles. The first proof defines the tangle fraction from the canonical form and uses flyping to prove in…
Study connects taffy pulling, fractions, and rational tangles.
problem Understanding the relationship between taffy pulling, fractions, and rational tangles.
method Developed a taffy analogue for Conway's characterization of rational tangles and gave a geometric connection.
result Direct geometric connection between rational tangles and taffy pulls.
New knots show linear independence in slice concordance.
problem Understanding the structure of rationally slice knots.
method Provided an infinite family of knots that are linearly independent.
result Found knots that are linearly independent and infinite order.
The paper proves that rational concordance of double twist knots is reciprocal.
problem Determining when double twist knots are rationally slice.
method Using Donaldson's diagonalization theorem, the paper constructs rational ball obstructions for non-rational sliceness.
result Infinitely many prime power-fold cyclic branched covers do not bound a rational ball, proving the converse of rational concordance.
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…
This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots (math.GT/0212011). The present paper includes a self-contained account of the tangl…
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.
problem Understanding when Lagrangian pinwheels embed in symplectic rational and ruled surfaces.
method Almost toric fibrations and symplectic rational blow-up.
result A rational homology ball embeds into a rational homology cylinder if and only if the parameter is greater than or equal to 1.
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
Study shows conditions for rational ellipticity of manifolds with symmetries.
problem Conditions for rational ellipticity of manifolds with symmetries.
method Analyzes conditions on compact simply connected manifolds with G-actions. result Proves rational ellipticity of M/G if M satisfies certain conditions.