Paper generalizes Kreweras triangle using universal sl_2 weight system.
problem Understanding finite order knot invariants.
method Defining a family of polynomials and showing their appearance in the universal sl_2 weight system.
result Polynomials generalize Kreweras triangle, refining normalized median Genocchi numbers.
Research disproves the extension of a weight system to a 4-invariant for graphs.
problem Whether the sl(2)-weight system extends to a unique 4-invariant of graphs.
method Analyzing the sl(2)-weight system and constructing recurrence relations for extensions.
result The sl(2)-weight system does not extend to a 4-invariant for graphs in full generality.
Constructs a 4-invariant for graphs at c = 3/8.
problem No specific problem stated; focuses on construction.
method Constructs a 4-invariant that extends a specialization of the sl(2)-weight system at c = 3/8, satisfying a deletion-contraction relation.
result Satisfies a simple deletion-contraction relation.
Formula for sl2 weight system on complete bipartite graphs.
problem Computing values of sl2 weight system for chord diagrams. method Chmutov-Varchenko recurrence relation, Hopf algebra projections.
result Computed values for chord diagrams with complete bipartite intersection graphs.
The purpose of this paper is twofold. On one hand, we introduce a modification of the dual canonical basis for invariant tensors of the 3-dimensional irreducible representation of Uq(sl2), given in terms of Jacobi diagrams, a central tool in quantum topology. On the other hand, we use this modified basis to study t…
We give a construction of Kirby weight systems associated to sl(2) and valued into the finite field Z/pZ. We show that it is possible to apply this sequence of weight systems on the universal invariant of framed link. We also show that the corresponding sequence admits a Fermat limit, which defines an asymptotic ration…
We introduce a new series Rk, k=2,3,4,…, of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small o…
The paper calculates a specific weight system for chord diagrams with a particular graph structure.
problem Calculating a specific weight system for chord diagrams with a complete bipartite graph structure.
method Using a Lie algebra sl3 and its weight system, the authors derive a function on chord diagrams. result The authors compute the sl3 weight system for chord diagrams with a complete bipartite graph structure. Develops method to construct Lie algebra weight system kernel using Vogel algebra.
problem Detecting correlators and distinguishing knots in 3D Chern-Simons theory.
method Uses Vogel's Λ algebra and Jacobi diagrams.
result Explicitly provides Jacobi diagrams in the kernel of sl_N weight system.
New weight systems derived from a specific Lie algebra for knot invariants.
problem Constructing universal weight systems for knot invariants.
method Using a minimal Z22-graded Lie algebra to create weight systems. result Weight system derived from A1ε shows hybrid properties of sl(2) and gl(1∣1). The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…
Graphs and local systems count multiwebs.
problem Counting multiwebs in graphs with local systems.
method Using Kasteleyn matrices and web-traces.
result Determinant of Kasteleyn matrix counts multiwebs.
The paper explores weight systems and their applications to graph and embedded graph invariants.
problem Developing weight systems for graphs and embedded graphs.
method Construction of weight systems from graph invariants and metrized Lie algebras, and extending to arbitrary embedded graphs.
result Explicit forms of generating functions and recurrence relations for weight systems on chord diagrams and embedded graphs.
The paper constructs bases for cluster varieties using mSL3-webs and laminations.
problem Cluster varieties associated to mSL3-local systems on surfaces. method Introducing mSL3-laminations, developing quantum and classical trace maps, and constructing bases. result Bases of regular functions on mPGL3 cluster varieties constructed from mSL3-laminations. One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his cla…
SL2MF predicts synthetic lethality using logistic matrix factorization.
problem Predicting synthetic lethality in human cancers from limited experimental data.
method Logistic matrix factorization incorporating biological knowledge.
result SL2MF effectively predicts known and unknown SL interactions.
Define web algebras for annular SL(2) and SL(3) using foam TQFTs.
problem Define invariant algebras for tangles in the annulus.
method Use annular foam TQFTs to assign complex of bimodules to tangles.
result Establish properties of algebras and bimodules, and provide a bijection between webs and paths.
Let S be a path-connected, locally-compact CW-complex, and let M be a subcomplex with finitely-many components. A `decorated SL_2(C)-local system' is an SL_2(C)-local system on S, together with a choice of `decoration' at each component of M (a section of the stalk of an associated vector bundle). We study the (decorat…
Let Vect(R) be the Lie algebra of smooth vector fields on R. The space of symbols Pol(T^* R) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(R)-module that becomes trivial once the action is restricted to sl(2). The deformations of Pol(T^* R), which become trivial once…
Novel symmetry found in colored HOMFLY polynomials from superalgebras.
problem Understanding symmetries in colored HOMFLY polynomials.
method Exploring the sl(N∣M) superalgebra to find a symmetry. result A symmetry relating polynomials colored by different representations.
Defines and parametrizes sl(2)-type singular fibres in symplectic and odd orthogonal Hitchin systems.
problem Characterizing and understanding singular fibres in Hitchin systems.
method Stratification by semi-abelian spectral data, study of irreducible components, global description of degenerations.
result Extension of Langlands duality to sl(2)-type Hitchin fibres. Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a defor…
New systems of linear PDEs discovered in 3D contact manifolds.
problem Investigating linear PDEs of sl3-type. method Complete local classification using extrinsic geometry.
result 7 new systems of second-order linear PDEs with 8-dimensional solution spaces.
Computes Lie algebra structure constants using a graphical calculus.
problem Computing Lie algebra structure constants efficiently.
method Graphical calculus for classical invariant theory.
result Generalizes known methods for sl2 to other Lie algebras. Proves generic independence and additivity of SL(2,C) Casson-Lin invariant.
problem SL(2,C) Casson-Lin invariant's parameter dependence and additivity under knot sums.
method Topology, microlocal analysis, algebraic geometry, Behrend functions.
result Generically independent and additive invariant under knot sums.
We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of Uq(sl2). We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of t…
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
Given an irreducible unitary representation of a cocompact lattice of SL(2,C), we explicitly write down a solution of the Strominger system of equations. These solutions satisfy the equation of motion, and the underlying holomorphic vector bundles are stable.
We give an explicit graded cellular basis of the sl3-web algebra KS. In order to do this, we identify Kuperberg's basis for the sl3-web space WS with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weigh…
The paper establishes isomorphisms and constructs colored versions of Lawrence representations.
problem Understanding isomorphisms and colored versions of Lawrence representations.
method Explicit isomorphisms and construction of colored versions.
result Matrices for colored versions of BKL and Lawrence representations provided.
The paper studies geometric structures on SL(n,R) induced by the Killing form.
problem Understanding geometric structures on SL(n,R) induced by the Killing form.
method Constructing manifolds, studying Poisson-commutation relations, and solving Hamiltonian systems.
result Explicit solutions of Hamiltonian systems for n=2.
A method for concise fuzzy system modeling using ESSC-SL-CTSK-FS.
problem Complex nonlinear systems with high-dimensional data and large numbers of rules.
method Integrating ESSC for antecedents and SL for consequent parameters optimization.
result Effective reduction in the number of fuzzy rules for clearer and more interpretable models.
We describe a minimal global coordinate system of order 30 on the SL(4,C)-character variety of a rank 2 free group. Using symmetry within this system, we obtain a smaller collection of 22 coordinates subject to 5 further real relations that determine conjugation classes of generic pairs of matrices in SU(3,1).
Study SL(2,C) connections on Seifert-fibered spaces using gauge theory.
problem Counting SL(2,C) connections on Seifert-fibered spaces. method Introduced perturbations of the SL(2,C) Chern--Simons functional and proved a localisation result. result Formulae for the Euler characteristic and Poincaré polynomial of the stable locus of the SL(2,C) character variety of a Seifert-fibered homology 3-sphere. A new method combines conformal prediction with Super Learner for interval predictions.
problem Constructing reliable interval predictions for complex regression functions.
method Coupling conformal prediction with Super Learner framework.
result The conformalized SL achieves valid finite-sample coverage with competitive performance.
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL2(Z) to its preimage in the universal cover of SL2(R). With this method we recover the classification of two-dimensional toric fans, and obtain a des…
New central elements found in a quantum algebra related to knot theory.
problem Understanding the algebraic structure of SLd-skein algebras. method Threaded polynomials from symmetric functions.
result Extraction of central elements in SLd-skein algebra. Describes spectral data for singular fibres of a specific Hitchin system.
problem Characterizing singular fibres of the SL(2,C)-Hitchin system. method Using Hecke transformations and analysis of parameter spaces, the paper stratifies and compactifies the singular spaces.
result Large classes of singular fibres are shown to be fibre bundles over Prym varieties.
This paper presents a construction of fibered links (K,Σ) out of chord diagrams $\sL$. Let Γ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of (K,Σ) equals the bilinear form of the simply-laced Coxeter system (W,S) associated to Γ; and the monodromy of $(K,Σ)…
Extends Lawrence's representations to integral Uqsl(2) Verma-modules and braid groups.
problem Integrating Lawrence's representations into Uqsl(2) Verma-modules and braid groups. method Defining homological operators and showing they provide a representation for Uqsl(2), establishing isomorphisms and preserving key properties. result Recovering an integral version of Kohno's theorem for Verma-modules and braid group representations.
The paper studies cohomological Donaldson-Thomas theory for local systems on a 3-torus.
problem Cohomological Donaldson-Thomas theory for local systems on the 3-torus.
method Using exponential maps and the tripled Jordan quiver, the paper proves cohomological integrality for GL_n and SL_n local systems.
result The paper proves Langlands duality statements for SL_n and PGL_n cohomological Donaldson-Thomas invariants for prime n.
Study on monodromy of cyclic opers on Riemann sphere with single pole.
problem Analyzing the monodromy of meromorphic cyclic opers on the Riemann sphere.
method Developed a method based on isomonodromic deformations and properties of structure constants for sl(n,C).
result Monodromy map is an immersion when the order of the pole is a multiple of n.
Unified theories for colored sl(2) knot homology.
problem Equivalence of different models for colored sl(2) knot homology.
method Conceptualized properties into a Chebyshev system and proved its uniqueness.
result Equivalence of Khovanov and Cooper-Krushkal models for the unknot.
The space of m-ary differential operators acting on weighted densities is a (m+1)-parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between this space and the corresponding space of symbols as sl(2)-modules. This yields to the notion o…
Twisted SL2C local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …
We introduce a quotient of the affine Temperley-Lieb category that encodes all weight-preserving linear maps between finite-dimensional sl(2)-representations. We study the diagrammatic idempotents that correspond to projections onto extremal weight spaces and find that they satisfy similar properties as Jones-Wenzl pro…
Rank inequality proven for annular Khovanov homology of 2-periodic links.
problem Proving a rank inequality for annular Khovanov homology of 2-periodic links.
method Spectral sequence analysis with quantum and sl2 weight spaces.
result Proven rank inequality rank AKhj,k(L)≤rank AKh2j−k,k(ildeL). Minimal crossing number found in arithmetic curve systems.
problem Finding the minimal crossing number in arithmetic curve systems.
method Analyzing systoles of hyperbolic surfaces associated with congruence lattices in SL2(Z).
result Minimal crossing number is asymptotically achieved.