Study of monopoles and q-difference modules correspondence.
problem Finding a correspondence between algebraic and differential geometric objects.
method Analogue of non-abelian Hodge theory for q-difference modules. result Doubly periodic monopoles and parabolic q-difference modules correspond. Introduces modular q-holonomic modules to solve q-difference equations.
problem Solving q-difference equations in quantum invariants and Chern-Simons theory. method Defines modular q-holonomic modules with improved analyticity properties. result Modular q-holonomic modules explain structural properties of quantum invariants and Chern-Simons theory. Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.
problem Understanding the genus-two handlebody skein module.
method Using q-difference operators for the genus-two skein algebra.
result Correspondence between reduced Askey-Wilson polynomials and genus-two handlebody skein module.
A sequence of rational functions in a variable q is q-holonomic if it satisfies a linear recursion with coefficients polynomials in q and qn. We prove that the degree of a q-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding hol…
Study of 3d-3d correspondence involving q-Weyl algebra and 3d-index.
problem Understanding the action of a q-Weyl algebra on the 3d-index of knots. method Investigation of the q-Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q-difference equations, and rational function matrix for the 3d-index determination. The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequence…
The paper calculates a formula for knot complements using holomorphic curves.
problem Calculating the partition function of knot complements.
method Skein valued holomorphic curve counting techniques.
result The partition function localizes on specific holomorphic annuli for torus knots.
In this paper we develop an asymptotic analysis for formal and actual solutions of q-difference equations, under a regularity assumption. In particular, evaluations of regular solutions of regular q-difference equations have an exponential growth rate which can be computed from the q-difference equation. The motivation…
Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the 7_4 knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot 7_4 with reducible non-abelian SL(2,C) character variety. To achieve our goal, we use symbolic summatio…
Quantum K-theory of quintic 3-fold conjectured with non-polynomial coefficients.
problem Reconstructing quantum K-theory for quintic 3-fold.
method Formulated explicit conjecture for small J-function and its q-difference equation.
result Coefficients of q-difference equations are non-polynomial functions of Gopakumar-Vafa invariants.
The paper defines a function for knots in Seifert manifolds and connects it to Witten-Reshetikhin-Turaev invariants.
problem Defining a function for knots in Seifert manifolds.
method Explicit construction of a function Φ(q; N) and its properties.
result The function Φ(q; N) satisfies a q-difference equation related to character varieties.
Complex Chern-Simons theory reveals peacock patterns in perturbative series.
problem Understanding the structure of partition functions in complex Chern-Simons theory.
method Analyzing the partition function as a holomorphic function and using resurgence theory.
result Perturbative series are resurgent, with trans-series involving non-perturbative variables.
Invariants of hyperbolic knots connect to quantum modularity.
problem Understanding quantum invariants of hyperbolic knots.
method Introducing matrix invariants and their properties.
result Matrix invariants relate to quantum modularity conjectures.
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and q-series to conjecture and compute knot invariants. result Explicit computations match conjectured values for specific knots.
We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, for instance, pairs of flat manifolds o…
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial A(x,y). Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted q or ℏ; this quan…
Classifies modules of surface-knots in terms of their properties.
problem Characterizing modules of surface-knots in terms of their properties.
method Using homology and covering spaces, the reduced first module is characterized.
result The reduced first module for every genus g is characterized in terms of properties of a finitely generated module.
New method learns both module structure and sequencing in neural networks.
problem Learning only the parameters and order of execution of neural modules.
method Expands the approach to learn the internal structure of modules, including the ordering and combination of arithmetic operators.
result Performance comparable to hand-designed modules achieved without extra supervisory signals.
Curvature defined for Hilbert modules and Kasparov modules.
problem Defining and studying curvature in Hilbert modules and Kasparov modules.
method Introduced curvature for densely defined universal connections on Hilbert C∗-modules relative to spectral triples. result Curvature only depends on the represented form of the universal connection modulo junk forms.
The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of …
Proves finiteness and holonomicity of skein modules for 3-manifolds.
problem Finiteness and holonomicity of skein modules for 3-manifolds.
method Defining skein transfer bimodules and using q-analogues of D-module theory.
result Internal skein modules are holonomic modules over the internal skein algebra of the boundary.
Defines super projective modules and explores their properties.
problem Exploring the geometric-algebraic link in super geometry.
method Defined and explored super projective modules over supersmooth functions.
result Module of vector fields over a supersphere is a super projective module.
A new method to derive presentations of skein modules is developed. For the case of homotopy skein modules it will be shown how the topology of a 3-manifold is reflected in the structure of the module. The freeness problem for q-homotopy skein modules is solved, and a natural skein module related to linking numbers is …
We introduce higher skein modules of links generalizing the Conway skein module. We show that these modules are closely connected to the HOMFLY polynomial.
Paper compares skein modules to Kauffman bracket modules.
problem Comparing skein modules to Kauffman bracket modules.
method Using skein relations and Reshetikhin-Turaev model.
result Resolved the problem of comparing skein modules to Kauffman bracket modules.
Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills of homologies and homotopies, distant mountains - the Kauffman bracket skein mo…
Generalized Steinberg module presentation for Gaussian and Eisenstein integers.
problem Presenting Steinberg modules for specific number rings.
method Generalization of Bykovskii's presentation to Gaussian and Eisenstein integers.
result Generalization does not yield a presentation for all Euclidean number rings.
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…
Combinatorial approach to compute satellite knot invariants using graph theory.
problem Computing knot invariants for satellite knots using bordered Heegaard Floer homology.
method Construct weighted A∞-modules using decorated planar graphs and prove their isomorphism. result Combinatorial proof of A∞ structure relations for the constructed modules. New sl(2) action defined on a mathematical module.
problem No specific problem stated; focuses on mathematical construction.
method Construction of sl(2)-action on equivariant skein lasagna module.
result Infinitesimal sl(2)-symmetries constructed.
Studies modules over a category of Jacobi diagrams in handlebodies.
problem Understanding modules over a specific category of Jacobi diagrams.
method Generalizes adjunctions and studies subquotient modules.
result Generalizes adjunctions between modules and Casimir Lie algebra modules.
Enhances knot and link invariants using quandle modules.
problem Distinguishing knots and links using polynomial invariants.
method Integrates quandle modules into the quandle coloring quiver.
result The enhanced invariant distinguishes knots and links.
Introduces admissible skein modules for non-semisimple categories.
problem No specific problem stated; generalization of Kauffman skein algebra.
method Introduces admissible skein modules associated to ideals in pivotal categories.
result These modules generalize Kauffman skein algebra and relate to quantum invariants.
Let {T1,…,Tn} be a set of n commuting bounded linear operators on a Hilbert space H. Then the n-tuple (T1,…,Tn) turns H into a module over C[z1,…,zn] in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \raro \clh, \quad \quad …
This paper generalizes L2 cohomology theory for complex manifolds.
problem Developing a L2 cohomology theory for Hodge modules on infinite covering spaces.
method Formulating a conjectural generalization of L2-Mixed Hodge structures using Saito's Mixed Hodge Modules.
result Partial results in the conjectural generalization of L2-Mixed Hodge structures.
Study Kauffman bracket skein modules of Seifert fibered spaces.
problem Understanding the structure of Kauffman bracket skein modules.
method Investigate spanning sets and module structure.
result Kauffman bracket skein modules are finitely generated.
Formula for interleaving distance of rectangle persistence modules.
problem Calculating distances between rectangle persistence modules.
method Formulas based on rectangle geometry, extended to decomposable modules.
result Closed formulas for interleaving and bottleneck distances.
Introduces Floer lasagna modules using link Floer homology.
problem No specific problem stated; focuses on new mathematical concept.
method Inspired by skein lasagna module, uses link Floer homology.
result Computes Floer lasagna modules for specific 4-manifolds.
New formula proves skein modules are finite for 3-manifolds.
problem Proving skein modules are finite for closed 3-manifolds.
method Using Heegaard splittings and algebraic computation.
result Skein modules are finite-dimensional, resolving a conjecture.
Consider the Chern-Simons topological quantum field theory with gauge group SU(2) and level k. Given a knot in the 3-sphere, this theory associates to the knot exterior an element in a vector space. We call this vector the knot state and study its asymptotic properties when the level is large. The latter vector space b…
Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
We show that the Kauffman bracket skein module of a cylinder over the torus embeds as a subalgebra of the noncommutative torus. Using this we derive nice formulas for the Jones-Wenzl idempotents and analyze the structure of the Kauffman bracket skein module of the unknot as a module over the Kauffman bracket skein modu…
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
The dualizing module of GL_n(O) varies, affecting cohomology vanishing and nonvanishing.
problem Understanding the dualizing module of GL_n(O) and its impact on cohomology.
method Analyzing the Steinberg module and a variant for GL_n(O), proving vanishing and nonvanishing theorems.
result The dualizing module of GL_n(O) is not always the Steinberg module, but a variant that accounts for orientation.
Study quandle modules over geometric quandles and their relation to Lie-Yamaguti representations.
problem Understanding quandle modules and their connection to Lie-Yamaguti representations.
method Examine quandle modules over quandle spaces, focusing on geometric structures.
result Modules over quandle spaces are linked to representations of Lie-Yamaguti algebras.
Explains connections between monopoles and modules on elliptic curves.
problem Understanding relationships between different mathematical objects.
method Explains equivalences between monopoles and polystable bundles and modules.
result Monopoles and polystable difference modules on elliptic curves are equivalent.
The multivariate Alexander module of a link L has several subsets that admit quandle operations defined using the module operations. One of them, the fundamental multivariate Alexander quandle, determines the link module sequence of L.