As a generalization of a fundamental result about the Alexander polynomial of links, we give a description of a Torres condition for the twisted Alexander polynomial of links associated to a unimodular representation.
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Conditions for integer signatures of high-dimensional knots.
problem Determining signatures of high-dimensional knots with specific Alexander polynomials.
method Necessary and sufficient conditions based on square-free Alexander polynomials.
result Identifies conditions for an integer to be the signature of a knot.
Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
Study on periodic knots, proving limitations on their Alexander polynomials.
problem Understanding Alexander polynomials of periodic knots.
method Polynomial factorization, number theory interpretation, computational methods.
result Alexander polynomials of freely periodic knots are restricted to products of cyclotomic polynomials.
Link signature limit depends on linking matrix under specific polynomial condition.
problem Limits of Tristam-Levine signature function under precise polynomial conditions.
method Analysis of Alexander polynomial and linking matrix.
result Limit of Tristam-Levine signature at 1 determined by linking matrix under specific polynomial condition.
It is a well known result from Thistlethwaite that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module, we describe an alt…
New methods assess topological entanglement in periodic systems.
problem Assessing topological entanglement in systems with periodic boundary conditions.
method Introducing Periodic Jones polynomial and Cell Jones polynomial.
result Periodic Jones polynomial is a recurring factor of Jones polynomial of finite cutoffs.
Ancient caloric functions on manifolds with polynomial growth are studied under volume doubling barrier.
problem Analyzing ancient caloric functions on manifolds beyond volume doubling.
method Time polynomial structure result on ancient caloric functions with polynomial growth.
result Finiteness result for ancient caloric functions is essentially sharp, except for multi-end cases.
This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.
Research on mixed polynomials, extending non-degeneracy concepts to complex variables.
problem Extending non-degeneracy concepts to mixed polynomials in complex variables.
method Generalization of Mondal's partial non-degeneracy to mixed polynomials, introducing new concepts and proving properties.
result Strong partial non-degeneracy implies isolated singularities, and mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition.
Tribracket Polynomialsmath.GT
Innovates polynomial invariant for tribrackets.
problem Counting and distinguishing knots and links.
method Introduces subtribracket polynomials and uses them to enhance counting.
result Enhanced counting invariant for knots and links.
Study Betti and Hodge numbers of solvmanifolds from integer polynomials.
problem Computing Betti and Hodge numbers of solvmanifolds constructed from integer polynomials.
method Analyzing de Rham and Dolbeault cohomology of solvmanifolds under algebraic conditions.
result Explicit generating polynomials for Hodge numbers in quasi full rank case.
We provide necessary conditions for the Alexander polynomials of algebraically split component-preservingly amphicheiral links. We raise a conjecture that the Alexander polynomial of an algebraically split component-preservingly amphicheiral link with even components is zero. Our necessary conditions and some examples …
Cappell-Shaneson polynomialsmath.GT
The paper defines and classifies Cappell-Shaneson polynomials.
problem Characterizing Cappell-Shaneson polynomials.
method Algebraic conditions on polynomials, reduction modulo primes, and construction of infinite series.
result Complete lists of Cappell-Shaneson polynomials of degrees 4 and 5, and several infinite series of degree 6.
Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including to knots with trivial Alexander polynomial, such as the two polynomial 1 knots w…
New framework uses score-based priors to solve ill-conditioned polynomial equations, improving signal recovery from noisy data.
problem Recovering signals from low-order moments in inverse problems, especially ill-conditioned polynomial equations.
method Integrates score-based diffusion priors with moment-based estimators to regularize and solve nonlinear inverse problems.
result Diffusion priors improve recovery from third-order moments and make super-resolution MTD feasible.
We determine prime amphicheiral links with at least 2 components and up to 11 crossings. There are 27 such links. We check also special amphicheiralities. Most of prime links with up to 11 crossings are detected not to be amphicheiral by a condition on the Jones polynomial. For the rest links, we applied conditions fro…
Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.
problem Classify bi-Lipschitz equivalence of mixed polynomials with inner non-degeneracy.
method Defined metric links and introduced new data to determine bi-Lipschitz equivalence.
result Neither Newton boundary nor C-face diagram is an invariant for bi-Lipschitz equivalence.
This study shows the moment-SOS hierarchy converges in polynomial optimization over product of spheres.
problem Minimizing multihomogeneous polynomials over product of spheres.
method Moment-SOS hierarchy, local optimality conditions, differential geometry, Morse theory.
result The moment-SOS hierarchy has finite convergence for generic multihomogeneous objective functions.
Polynomial Jump-Diffusion Modelsq-fin.MF
We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is preserved under polynomial transformations and Lévy time change. We present a generic method for…
The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops …
For knots in , it is well-known that the Alexander polynomial of a ribbon knot factorizes as for some polynomial . By contrast, the Alexander polynomial of a ribbon -knot is not even symmetric in general. Via an alternative notion of ribbon -knots, we give a topological condition on a $…
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
Alexander polynomial condition blocks crossing changes in some knots.
problem Cosmetic crossing changes in knot diagrams.
method Alexander polynomial obstruction for -space knots.
result Proved the conjecture for a five-parameter family of pretzel knots.
New bound on Jones polynomial for specific positive links.
problem Finding bounds on the Jones polynomial for positive links.
method Using previous results on positive fibered links, we found a new bound for a specific family of positive links.
result We provided a bound on the maximum degree of the Jones polynomial for positive links with a specific coefficient.
The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.
problem Classifying pretzel links based on their self delta-equivalence.
method Using Conway polynomials to determine self delta-equivalence for links with 2 or more components.
result Necessary and sufficient conditions for self delta-equivalence of pretzel links with 3 or more components.
Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
problem Analyzing locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.
method Examines Riemannian and Finslerian surfaces, providing necessary and sufficient conditions for locally symmetric fourth root metrics in 2D and more complex conditions for higher dimensions.
result Formulates conditions for positive definiteness of locally symmetric polynomial metrics in Finslerian surfaces and provides explicit examples.
New estimator adapts to various error distributions.
problem Adapting to different error distributions in nonparametric regression.
method Introduces outrigger local polynomial estimator with modified weighted least squares.
result Minimax optimal over Hölder classes with multiplicative factor.
We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using …
The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.
problem Identifying knots with nontrivial Alexander polynomials and improving knot classification.
method Elimination of infinite families of knots and use of determinants to improve unknotting number data.
result Elimination of infinite families of knots with nontrivial Alexander polynomials and improvement of unknotting number data.
This paper simplifies conditional Sobol' indices calculation using PCE bases.
problem Computational inefficiency and lack of consistency in evaluating conditional Sobol' indices.
method Analytical extraction of conditional Sobol' indices via basis decomposition of PCE expansions.
result Derives closed-form expressions for conditional Sobol' indices.
Third coefficient of lens space knots' Alexander polynomials restricts surgeries to specific torus knots.
problem Restricting lens space surgeries to specific configurations.
method Analyzing Alexander polynomials of lens space knots and their surgeries.
result Third coefficient condition confines surgeries to -torus knots.
By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
The paper simplifies the computation of a complex polynomial using Yang-Baxter operators.
problem Computing the homology of Yang-Baxter operators for arbitrary m.
method Reduced the computation to initial conditions and produced explicit formulas.
result Explicit formulas for the third and fourth homology.
We give necessary conditions for a polynomial to be the Conway polynomial of a two-bridge link. As a consequence, we obtain simple proofs of the classical theorems of Murasugi and Hartley. We give a modulo 2 congruence for links, which implies the classical modulo 2 Murasugi congruence for knots. We also give sharp bou…
Let be the fundamental group of the exterior of a knot in the three-sphere. We study deformations of representations of into which are the sum of two irreducible representations. For such representations we give a necessary condition, in terms of the twisted Alexander polynomial, for…
Polynomial-time algorithm learns ReLU networks without assumptions.
problem Learning linear combinations of ReLU activations with Gaussian inputs.
method Random contractions of moment tensors and multi-scale analysis.
result First polynomial-time algorithm without additional assumptions.
In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree.
The taut polynomial equals a twisted Alexander polynomial.
problem Understanding the relationship between taut polynomials and Alexander polynomials.
method Defined taut polynomial of veering triangulations and proved it equals a twisted Alexander polynomial.
result The taut polynomial equals a twisted Alexander polynomial of the underlying manifold.
We introduce polynomial processes taking values in an arbitrary Banach space via their infinitesimal generator and the associated martingale problem. We obtain two representations of the (conditional) moments in terms of solutions of a system of ODEs on the truncated tensor algebra of dual respectively bidual s…
In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, t…
The paper studies conditions for graphs connecting level sets of harmonic polynomials.
problem Conditions for graphs connecting level sets of harmonic polynomials.
method Algebraic properties and Kempf-Ness functional construction.
result Stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation.
We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomial…
Polynomial processes have the property that expectations of polynomial functions (of degree , say) of the future state of the process conditional on the current state are given by polynomials (of degree ) of the current state. Here we explore the application of polynomial processes in the context of structur…
Study polynomial structures on generalized tangent bundles and their compatibility with operators.
problem Understanding polynomial structures on generalized tangent bundles.
method Analyzing skew-symmetric endomorphisms satisfying polynomial equations and their compatibility with de Rham and Courant-Dorfman operators.
result Conditions equivalent to integrability of generalized almost complex structures.
In this paper, we study harmonic and caloric functions of polynomial growth on a complete non-compact gradient shrinking Ricci soliton. On one hand, when the scalar curvature satisfies at least quadratic decay, we prove that the space of harmonic functions with fixed polynomial growth degree is finite dimensional. We a…
Holonomy-preserving transformations help recover Alexander polynomials from graph zeta functions.
problem Recovering Alexander polynomials from graph zeta functions.
method Introducing holonomy to preserve zeta functions of matrix-weighted graphs and extending to group elements and quandles.
result Holonomy-preserving transformations correspond to transformations of group presentations and preserve the twisted Alexander polynomial.
New results on algebraic knots with Brieskorn polynomials.
problem Understanding cobordisms of algebraic knots defined by Brieskorn polynomials.
method Analyzing Fox--Milnor type relations, decomposing algebraic cobordism classes, and studying cyclic suspensions.
result Spherical algebraic knots associated with Brieskorn polynomials have infinite order in the knot cobordism group.