Minimal polynomial found for Riemannian C_0-spaces.
problem Understanding the structure of Riemannian C_0-spaces.
method Constructing polynomial functions on tangent spaces and gluing them globally.
result The degree of the polynomial provides an upper bound for the Singer invariant.
Sharp upper bound for quasi polynomial degree of manifold configuration spaces.
problem Determining the exact degree of quasi-polynomial homology groups of configuration spaces.
method Analyzing extremal homology groups of unordered configuration spaces of manifolds.
result The upper bound for the degree of quasi-polynomials is sharp for every manifold.
The study classifies polynomial relation tubular surfaces in 3-spaces.
problem Classifying tubular surfaces with polynomial curvature relations.
method Analyzing polynomial relations between Gaussian and mean curvatures in Euclidean, hyperbolic, and Lorentzian 3-spaces.
result Determination of sets of polynomial relations for tubular surfaces.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
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.
We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fibre. It follows from these relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in len…
Novel Jones polynomial for open curves in 3D space.
problem Measuring entanglement complexity of open curves in 3-space.
method Defining Jones polynomial for linkoids and extending to collections of open and closed curves.
result Jones polynomial for open curves has real coefficients and is continuous.
New Frobenius manifold structures found on Dicyclic group orbits.
problem Finding Frobenius manifold structures on orbits spaces of Dicyclic groups.
method Applying Dubrovin's method to Dicyclic groups.
result Dicyclic group orbits spaces acquire two Frobenius manifold structures.
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 (2,2g+1)-torus knots. Spaces of polynomials are shown to be Euclidean balls.
problem Understanding the geometry of Lorentzian and real stable polynomials.
method Refined connection between symmetric exclusion process and polynomial geometry.
result Spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls.
The paper calculates the Hilbert polynomials for configuration spaces over graphs with a short circumference.
problem Calculating the Betti numbers of configuration spaces over graphs with a short circumference.
method Using a combinatorial approach based on the canonical 1-bridge decomposition of the graph.
result An expression for the Hilbert polynomial of a graph in terms of its canonical 1-bridge decomposition.
Homology groups of spaces of nonsingular polynomial embeddings R1→Rn of degrees ≤4 are calculated. A general algebraic technique of such calculations for spaces of polynomial knots of arbitrary degrees is described.
New proof of Alexander polynomial constraints for lens space surgeries.
problem Constraints on Alexander polynomials for lens space surgeries.
method Using changemaker lattices to prove a theorem.
result Constraints on Alexander polynomials for specific surgeries.
Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type Dm. Based on this geometric interpretation he conjectured that these polynomials…
A polynomial knot in Rn is a smooth embedding of R in Rn such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space P of polynomial knots in R3 with the inductive limit topology coming from the spaces $\m…
Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's p…
Ancient solutions to biharmonic heat equation bounded by polynomial dimensions.
problem Bounding ancient solutions to biharmonic heat equation.
method Using polynomial volume growth and dimensions of biharmonic functions.
result Ancient solutions are bounded by polynomial dimensions.
New knots share same Upsilon invariant despite different Alexander polynomials.
problem Identifying concordant knots via Upsilon invariant.
method Examined hyperbolic L-space knots and their Upsilon invariants.
result Infinitely many pairs of hyperbolic L-space knots with distinct Alexander polynomials share the same Upsilon invariant.
Formula for Alexander polynomial of links with twists.
problem Computing Alexander polynomial of links with twists.
method Using vector space representation of Uq(gl(1∣1)). result Alexander polynomials stabilize after adding enough twists.
A simplified proof of the Alexander-Conway polynomial exists.
problem Existence of the Alexander-Conway polynomial for links in 3D space.
method Presented an accurate detailed exposition of the proof.
result Existence of the Alexander-Conway polynomial proved.
Globalizes Jones and Alexander polynomials using topological intersections.
problem Link invariants from graded intersections of Lagrangians.
method Topological model proving the Jones polynomial's well-definedness and constructing globalizations.
result Proves the Jones polynomial and constructs globalizations of Jones and Alexander polynomials.
We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous …
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
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.
The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. The subspace of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a free Z^d-action by automorphisms, C is a finitely generated module over the polynomial ring…
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. Study polynomial growth harmonic functions on infinite penny graphs.
problem Finite-dimensional property of polynomial growth harmonic functions on infinite penny graphs.
method Asymptotically sharp dimensional estimate for ancient solutions of the heat equation.
result Proved the asymptotically sharp dimensional estimate.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for Uq(sl(2)). method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
Coloured Jones and Alexander polynomials are sequences of quantum invariants recovering the Jones and Alexander polynomials at the first terms. We show that they can be seen conceptually in the same manner, using topological tools, as intersection pairings in covering spaces between explicit homology classes given by L…
Computes minimal polynomials for generalized Heisenberg groups.
problem None explicitly stated; focus on method.
method Computes minimal polynomials for generalized Heisenberg groups.
result Explicit minimal polynomials for generalized Heisenberg groups.
Study on moduli spaces of Higgs bundles over Abelian varieties, focusing on their topology and polynomials.
problem Determine the topology and polynomials of moduli spaces of Higgs bundles over Abelian varieties.
method Analyzing the Poincaré polynomials and mixed Hodge polynomials of moduli spaces MAH(G) for various groups G and dimensions d. result Explicit formulas for Poincaré polynomials and mixed Hodge polynomials in specific cases, including rank 2 and 3 Higgs bundles.
New Calabi-Yau metrics converge polynomially to Calabi model space.
problem Finding complete Calabi-Yau metrics with polynomial convergence rate.
method Defined new metrics on Calabi-Yau complements with ample normal bundles.
result Uniqueness of these metrics within a cohomology class.
New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.
problem Constructing a 3-manifold invariant from Viro's gl(1|1) polynomial.
method Following Costantino, Geer, and Patureau-Mirand's method in relative G-modular categories.
result The invariant can distinguish homotopy equivalent lens spaces.
Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.
problem Characterize geodesics in jet space and identify those that are globally minimizing.
method Sub-Riemannian geometry, Hamilton-Jacobi equations, and analysis of period degenerations.
result Some polynomials yield globally minimizing geodesics, with conjectures on the independence of cut time.
New formulas for knot polynomial evaluations from covering spaces.
problem Evaluating knot polynomials uniquely from covering spaces.
method Using singular determinants and linking pairings.
result Explicit formulae for Jones and Q-polynomial evaluations. New knot models analyze local entanglement for robust curve analysis.
problem Lack of local structural information in classical knot theory.
method Proposed multiscale and persistent Jones polynomials.
result Models are stable to small perturbations, robust for real-world applications.
Study on polynomial growth functions and forms on gradient Ricci solitons.
problem Estimating dimensions of polynomial growth holomorphic functions and forms.
method Relating to spectral data of the f-Laplacian, proving estimates under curvature assumptions. result Sharp dimension estimates and almost sharp frequency estimates for polynomial growth holomorphic functions.
In this paper, we study the quantum sl(n) representation category using the web space. Specially, we extend sl(n) web space for n≥4 as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial Pn(q) specialized to a one variable polynomial …
Jones polynomials compute weighted sums of Lefschetz numbers.
problem Computing Lefschetz numbers for braids.
method Colored Jones polynomials of braid closures.
result Jones polynomials compute abelianized Lefschetz numbers.
Study slice-regular polynomial functions via twistor space group actions.
problem Characterize slice-regular functions and their polynomial subclasses.
method Employ the twistor construction and group actions of PGL(2,H). result Characterize slice-regular functions with planar twistor lifts and normal classes of polynomials.
We construct a 2-variable link polynomial, called WL, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two 1-variable polynomials, one of which is the Alexander polynomial. We refine WL…
The paper studies the n-loop Kontsevich invariant of knots with same Alexander polynomial.
problem Understanding the n-loop Kontsevich invariant for knots with identical Alexander polynomials.
method Analyzes the subspace generated by the n-loop Kontsevich invariant of knots with genus ≤ g and same Alexander polynomial.
result For n ≥ 2, the subspace is finite-dimensional.
Holomorphic actions on complex spaces for nilpotent groups.
problem Understanding polynomial actions on complex spaces for nilpotent groups.
method Explicit construction of biholomorphisms by polynomial maps.
result Simply connected nilpotent Lie groups are biholomorphic to Cn. We introduce polynomial processes taking values in an arbitrary Banach space B via their infinitesimal generator L 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…
Let L be a oriented link such that Σn(L), the n-fold cyclic cover of S3 branched over L, is an L-space for some n≥2. We show that if either L is a strongly quasipositive link other than one with Alexander polynomial a multiple of (t−1)2g(L)+(∣L∣−1), or L is a quasipositive link other than …
We prove the existence of a polynomial invariant that satisfies the HOMFLY skein relation for links in a lens space. In the process we also develop a skein theory of toroidal grid diagrams in a lens space.
The paper studies geometric structures of polynomial spaces.
problem Understanding the geometric and combinatorial structures of polynomial spaces.
method Introducing and analyzing finite piecewise Euclidean cell complexes.
result The branched rectangle and annulus complexes are homeomorphic to specific polynomial spaces.
Study local moduli of Sasaki-Einstein metrics on specific polynomial links.
problem Understanding the local moduli of Sasaki-Einstein metrics on links of invertible polynomials.
method Analyzing Sasaki-Einstein metrics on links of invertible polynomials of cycle type and Thom-Sebastiani sums.
result For polynomials of cycle type, local moduli spaces are zero-dimensional. For Thom-Sebastiani sums, dimensions are positive.