Study calculates Kashaev invariants for twice-iterated torus knots.
problem Calculating Kashaev invariants for complex knots.
method Asymptotic analysis and topological interpretations.
result Obtained formulas linking invariants to Chern-Simons and Reidemeister torsion.
A new polynomial invariant for links in a thickened torus exhibits volume conjecture behavior.
problem Defining and characterizing a new polynomial invariant for links in a thickened torus.
method Defining a new invariant JnT, proving properties, and providing constructions. result The invariant JnT exhibits volume conjecture behavior, providing the first example of this in a virtual link. Study on knot polynomial's asymptotic behavior for figure eight.
problem Investigating the asymptotic behavior of colored HOMFLY polynomial for figure eight knot.
method Establishing an asymptotic expansion for the colored HOMFLY polynomial.
result Showed that Chern-Simons invariants and twisted Reidemeister torsion can be derived from the polynomial.
Study continuity of phi-invariant for degenerating graphs.
problem Continuity of phi-invariant for degenerating graphs.
method Use Yuan--Zhang's adelic divisors and follow Yuan's globalization of phi-invariants.
result Asymptotic expression of Zhang--Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space.
The paper analyzes the asymptotic behavior of a knot polynomial for a specific real number.
problem Understanding the asymptotic behavior of a knot polynomial for a real number.
method Examining the asymptotic behavior of the N-dimensional colored Jones polynomial evaluated at exp(ξ/N) for a real number ξ. result From the asymptotic behavior, the mSL(2;C) Chern--Simons invariant and the Reidemeister torsion twisted by the adjoint action can be extracted. Study on asymptotic behavior of knot invariants for figure eight knot.
problem Investigate asymptotic behavior of colored Jones polynomials and Turaev-Viro invariants for figure eight knot.
method Considered M-th colored Jones polynomials and Turaev-Viro invariants for figure eight knot with fixed limiting ratio s of M and (N+1/2). Found asymptotic expansion formula for colored Jones polynomials and showed exponential growth rate difference for s close to 1/2 and 1. Related Turaev-Viro invariants to colored Jones polynomials. result Asymptotic expansion formula for colored Jones polynomials and Turaev-Viro invariants of figure eight knot.
This paper uses SDEs to analyze GANs training and long-run behavior.
problem Understanding the training process and long-run behavior of GANs.
method Established SDE approximations for GANs training and analyzed long-run behavior via invariant measures.
result The long-run behavior of GANs training can be studied via the invariant measures of its SDE approximations.
We make policy optimization algorithms batch size-invariant by decoupling proximal and behavior policies.
problem Some policy optimization algorithms do not have batch size-invariance, leading to inefficiencies.
method We decouple the proximal policy from the behavior policy to achieve batch size-invariance.
result Our approach makes policy optimization algorithms more efficient and allows them to use stale data more effectively.
Study contact manifold heat kernels under Riemannian metrics blow-up.
problem Analyze spectral invariants on contact manifolds.
method Examine Hodge Laplacian heat kernel behavior under Riemannian metrics.
result Contact versions of eta-invariant and analytic torsion are topological.
We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the a…
Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.
problem Volume conjecture for Seifert fibered and graph 3-manifolds.
method Large r asymptotic behavior of Turaev-Viro invariants under gluing operation. result Volume conjecture proven for Seifert fibered and graph 3-manifolds.
Generative models synthesize microstructures respecting physical invariances.
problem Synthesizing microstructures given limited images and physical constraints.
method Three generative models: WGAN, physics-informed GAN, and hybrid model.
result Synthesized microstructures respect physical invariances and latent variable constraints.
The paper connects ADO polynomials to Vassiliev invariants for knots.
problem Connecting ADO polynomials to Vassiliev invariants for knots.
method Exploiting the colored Jones polynomials and their decomposition as Vassiliev invariants, the authors transpose this to ADO polynomials.
result A unique computable expansion of ADO polynomials as Vassiliev invariants.
After analyzing renormalization schemes on a Poincaré-Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well-known, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior und…
Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang invariant is as a quotient of two natural hermitian metrics with the same first Chern form on the line bundle of holomorphic differentials. In this paper we determine…
Paper compares two equivariant η-forms, revealing their singular behavior.
problem Comparing two equivariant η-forms to understand their singular behavior.
method Defined and compared equivariant infinitesimal η-form with equivariant η-form modulo exact forms.
result Obtained the singular behavior of the equivariant η-form as a function on the acting Lie group.
We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.
Motivated by conjectures in Heegaard Floer homology, we introduce an invariant HC(Y) of the cohomology ring of a closed 3-manifold Y whose behavior mimics that of the Heegaard Floer homology HF^\infty(Y,s) for s a torsion spin-c structure. We derive from this a numerical invariant h(Y), and obtain upper and lower bound…
The paper proposes a method to learn user representations invariant to social media behavior changes.
problem Difficulty in comparing users over time due to evolving behavior.
method Learning a mapping from user activity to a vector space capturing invariant features.
result The learned mapping enables efficient comparisons of users not seen at training time.
Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
problem Understanding the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Monotonicity of eigenvalues of mean curvature, convergence to geometric invariants.
result Eigenvalues of mean curvature converge to geometric invariants in the Gauduchon case.
Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.
problem Behavior of knot invariant θ under satellite operations.
method Proved additivity, introduced computational tool, verified conjecture for 2977 knots.
result Distinguished knots using invariant θ and verified conjecture for 2977 primes.
Study geometric properties of cuspidal edges with boundary.
problem Differential geometric properties of cuspidal edges with boundary.
method Analysis of differential geometric invariants and their relations.
result Relation between boundary behavior and other invariants.
Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.
problem Asymptotic behavior of colored Jones polynomials of figure-eight knot.
method Analyzes growth rate of polynomial evaluated at specific points.
result Growth rate determined by Chern-Simons invariant of an affine representation.
New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.
We make use of the action of H1(Y) in Heegaard Floer homology to generalize the Ozsváth-Szabó correction terms for 3-manifolds with standard HF∞. We establish the basic properties of these invariants: conjugation invariance, behavior under orientation reversal, additivity, and spinc ratio…
Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, an…
New filling functions for groups with coefficients show different asymptotic behavior.
problem Difficulty in filling loops with surfaces in Cayley graphs.
method Defining homological filling functions with coefficients and proving their differences.
result Filling functions for n-cycles with coefficients in different groups have distinct asymptotic behavior. We calculate the Witte-Reshetikhi-Turaev invariant for a knot in the lens space of type L(m,1) for the N-th root of unity, and study its asymptotic behavior for large N.
Study uses big data to analyze quantum invariants.
problem Investigate structural properties of Jones polynomial.
method Exploratory and topological data analysis, including coloring, rank increase, categorification.
result Contrasts behavior of Jones polynomial under various enhancements.
The paper examines the initial geometry of vacuum cosmological spacetimes and introduces new methods to characterize their behavior.
problem Characterizing the initial geometry of vacuum cosmological spacetimes.
method Analyzes Gowdy and non-Gowdy spacetimes with TN-actions, introduces a monotonic quantity for Kasner spacetimes, and uses curvature and volume bounds. result Evidence of AVTD behavior in Gowdy spacetimes and sufficient conditions for nonGowdy spacetimes.
We begin an exploration of parametric Backlund transformations for hyperbolic Monge-Ampere systems. We compute invariants for such transformations and explore the behavior of four examples regarding their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for furthe…
The paper explores invariant vs non-invariant complex structures on Lie groups.
problem Understanding complex structures on Lie groups and their properties.
method Analysis of invariant and non-invariant almost complex structures on compact quotients of Lie groups.
result New computations of Kodaira dimension for invariant and non-invariant structures.
We study the behavior of the Ozsvath-Szabo and Rasmussen knot concordance invariants tau and s on K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus kn…
I follow Y. Yokota to explain how to obtain a tetrahedron decomposition of the complement of a hyperbolic knot and compare it with the asymptotic behavior of Kashaev's link invariant using the figure-eight knot as an example.
We provide a proof and analyze the asymptotic behavior of a formula for the linking number of line segments.
problem The invariant formula for the linking number of line segments and its asymptotic behavior.
method Detailed proof and asymptotic analysis of the formula.
result We provide a proof and analyze the asymptotic behavior of the formula for the linking number of line segments.
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
Study differential invariants for Kundt waves, solving equivalence problem.
problem Equivalence problem for Kundt waves metrics.
method Scalar differential invariants with respect to the equivalence pseudo-group.
result Finitely represented algebra of invariants on the generic stratum.
Extends BCOV invariant to pairs of Calabi-Yau manifolds and pluricanonical divisors.
problem Extend BCOV invariant to new geometric pairs.
method Extend BCOV invariant to pairs (X,D), study blow-up behavior. result Results imply birational Calabi-Yau manifolds have the same BCOV invariant.
We give explicit formulæ for Noether invariants associated to Killing vector fields for the variational problem of minimal and constant mean curvature surfaces in 3-manifolds. In the case of homogeneous spaces, such invariants are the flux (associated to translations) and the torque (associated to rotations). Then we f…
Deviation from universal exponents in currency markets reveals economic properties.
problem Identifying invariant behavior in heterogeneous complex systems.
method Analysis of exchange rate dynamics in international currency market.
result Exponents of fluctuation distributions for different currencies diverge from a universal form.
I calculate optimistically asymptotic behaviors of the WRT SU(2) invariants for the three-manifolds obtained from the figure-eight knot by p-surgeries with p=0,1,2,...,10, from which one can extract volumes and the Chern-Simons invariants of these closed manifolds. I conjecture that this also holds for general closed t…
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
Study of Khovanov homology invariants from U(1)imesU(1)-equivariant algebra.
problem Understanding concordance invariants from Khovanov homology.
method Analysis of U(1)imesU(1)-equivariant Khovanov homology using algebraic filtrations. result Extracted two families of concordance invariants.
SAM improves generalization in overparameterized models, but its behavior in tensorized models is less understood.
problem Understanding the implicit regularization of SAM in tensorized models.
method Scale-invariance analysis and gradient flow analysis to derive Norm Deviation as a measure of core norm imbalance, and propose Deviation-Aware Scaling (DAS).
result DAS achieves competitive or improved performance over SAM, while offering reduced computational overhead.
We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We pr…
Study on quantum invariants from surgeries on torus knots.
problem Quantum invariants of three-manifolds from surgeries along torus knots.
method Analysis of Witten-Reshetikhin-Turaev invariant and Chern-Simons invariants.
result Quantum invariants can be described as sums of Chern-Simons invariants and twisted Reidemeister torsions.
We define a concordance invariant, epsilon(K), associated to the knot Floer complex of K, and give a formula for the Ozsváth-Szabó concordance invariant tau of K_{p,q}, the (p,q)-cable of a knot K, in terms of p, q, tau(K), and epsilon(K). We also describe the behavior of epsilon under cabling, allowing one to compute …
Study shows quantum modularity in figure-eight knot's colored Jones polynomial.
problem Asymptotic behavior of colored Jones polynomial of figure-eight knot.
method Analyzing polynomial evaluated at specific points and showing asymptotic equivalence.
result Quantum modularity demonstrated in the figure-eight knot's colored Jones polynomial.