An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, -genus and -genus of a positive knot are equal. In this paper, we p…
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48 results for “positive invariants”
Study of new link types and their invariants, extending previous results.
problem Properties of polynomial invariants and signatures of weakly successively almost positive links.
method Analysis of minimal genus and fibering properties, extending known theorems.
result Extension of Scharlemann-Thompson's theorem to weakly successively almost positive links.
We characterize positive links in terms of strong quasipositivity, homogeneity and the value of Rasmussen, Beliakova and Wehrli's -invariant. We also study almost positive links, in particular, determine the -invariants of almost positive links. This result suggests that all almost positive links might be strongl…
Study finds metrics with positive intermediate Ricci curvature on specific low-dimensional manifolds.
problem Existence of invariant metrics with positive intermediate Ricci curvature on low-dimensional cohomogeneity one manifolds.
method Construction of invariant metrics with positive intermediate Ricci curvature on specific manifolds.
result Invariant metrics with positive 4th-intermediate Ricci curvature exist but not for 3rd-intermediate Ricci curvature on certain manifolds.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac …
Study on quantum invariant for positive links.
problem Characterizing and understanding the quantum invariant of positive links.
method Skein theory of -webs, explicit formulae, diagrammatic quantities, obstructions.
result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…
In his study of Ricci flow, Perelman introduced a smooth-manifold invariant called lambda-bar. We show here that, for completely elementary reasons, this invariant simply equals the Yamabe invariant, alias the sigma constant, whenever the latter is non-positive. On the other hand, the Perelman invariant just equals + i…
We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.
The study establishes conditions for positive and quasi-positive links.
problem Characterizing and testing positive and quasi-positive links.
method Proves necessary conditions for link concordance and positivity.
result Characterizes positive links with unlinking number 1 and 2, and tests positive links as closures of positive braids.
Positive braids linked to knot invariants and geometric monodromy groups.
problem Understanding knot invariants and geometric monodromy groups for positive braids.
method Associate braid monodromy groups to positive braids, identify these groups with framed mapping class groups for knots, and use these to determine knot invariants.
result Geometric monodromy groups of irreducible singularities are determined by genus and Arf invariant of associated knots.
We define a new notion of thin position for a graph in a 3-manifold which combines the ideas of thin position for manifolds first originated by Scharlemann and Thompson with the idea of thin position for knots first originated by Gabai. This thin position has the property that connect summing annuli and pairs-of-pants …
Extends LOSS invariant naturality to positive contact surgeries.
problem Naturality of LOSS invariant under positive contact surgeries.
method Proves naturality of LOSS invariant for Legendrian knots under positive integer contact surgeries.
result Contact invariant of +n surgery depends only on classical invariants of S.
We compute the Dijkgraaf-Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf-Witten invariants of surfaces of positive genus are positive integers.
Study on metrics with positive scalar curvature on manifolds with singularities.
problem Understanding metrics with positive scalar curvature on manifolds with singularities.
method Proving homotopy invariance of the space of metrics with positive scalar curvature on manifolds with fibred singularities.
result Proved that the space of metrics with positive scalar curvature is homotopy invariant under certain surgeries.
Relative Yamabe Invariantmath.DG
We define a relative Yamabe invariant of a smooth manifold with given conformal class on its boundary. In the case of empty boundary the invariant coincides with the classic Yamabe invariant. We develop approximation technique which leads to gluing theorems of two manifolds along their boundaries for the relative Yamab…
For a Riemannian manifold with dimension at least six, we prove that the existence of a conformal metric with positive scalar and Q curvature is equivalent to the positivity of both the Yamabe invariant and the Paneitz operator.
Conformal invariants of manifolds of non-positive scalar curvature are studied in association with growth in volume and fundamental group.
We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G invariant Riemannian metric with positive sectional curvature. For the exceptions,…
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Optimal pinching results on Einstein manifolds with positive Yamabe invariant.
problem Understanding the rigidity of Einstein manifolds with positive Yamabe invariant.
method Optimal pinching results and bounds on scalar curvature and Weyl tensor norms.
result Improved bounds on the Yamabe invariant and scalar curvature.
The paper studies invariant metrics with positive scalar curvature on 3-manifolds.
problem Classifying -invariant 3-manifolds with positive scalar curvature.
method Analyzes the space of -invariant Riemannian metrics with positive scalar curvature on closed 3-manifolds.
result The space of -invariant PSC metrics is either empty or contractible.
New method produces positive colored superpolynomials from four-point functions.
problem Constructing positive superpolynomials for colored link invariants.
method Four-point functions from topological vertices.
result New relation between super- and hyperpolynomials.
The Kreck-Stolz -invariant is a classic path-component invariant for the space and moduli space of positive scalar curvature metrics. It is an absolute (as opposed to relative) invariant, but this strength comes at the expense of being defined only under restrictive topological conditions. The aim of this paper is t…
New distances defined on Legendrian spaces without positive loops.
problem Defining distances on Legendrian spaces without positive loops.
method Constructing unbounded invariant distances on Legendrian isotopy classes.
result Invariant distances on Legendrian isotopy classes are discrete.
Characterizes invariant spinors on flag manifolds.
problem Existence of non-trivial invariant spinors on flag manifolds.
method Based on combinatorial properties of positive roots.
result Bounds for the dimension of invariant spinors.
We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant …
We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are non-trivial.
Proves curvature positivity of invariant direct images in complex geometry.
problem Curvature positivity of invariant direct images in complex geometry.
method Compact group action and Hörmander's theory of .
result Direct image of Nakano positive vector bundle is Nakano positive.
The paper explores fibered and quasi-positive links, introducing new families and invariants.
problem Understanding the structure of -space links and their properties.
method Using the H-function as a concordance link invariant.
result Introduced a subfamily of fibered strongly quasi-positive -space links and an infinite family of non-quasi-positive -space links.
A homogeneous knot is a generalization of alternating knots and positive knots. We determine the Rasmussen invariant of a homogeneous knot. This is a new class of knots such that the Rasmussen invariant is explicitly described in terms of its diagrams. As a corollary, we obtain some characterizations of a positive knot…
STRING improves 2D and 3D position encodings for better performance.
problem Efficient and accurate position encoding for 2D and 3D applications.
method STRING extends Rotary Position Encodings with a unifying theoretical framework, maintaining translation invariance and low computational cost.
result STRING shows substantial gains in open-vocabulary object detection and robotics.
Study determines scalar curvature invariants for 3-spheres embedded in 4-manifolds.
problem Positive scalar curvature metrics on specific 4-manifolds.
method Relative Bauer-Furuta-type invariant on periodic-end 4-manifolds.
result Obstructions to positive scalar curvature metrics on rational homology .
Study verifies Homogeneity Conjecture for three odd-dimensional spheres in positive curvature.
problem Verifying the Homogeneity Conjecture for three specific odd-dimensional spheres in positive curvature.
method Developed methods to verify the conjecture for three odd-dimensional spheres.
result Completes verification of the Homogeneity Conjecture in positive curvature.
The second H. Weyl curvature invariant of a Riemannian manifold, denoted , is the second curvature invariant which appears in the well known tube formula of H. Weyl. It coincides with the Gauss-Bonnet integrand in dimension 4. A crucial property of is that it is nonnegative for Einstein manifolds, hence it p…
Study eta invariant on non-compact manifolds with positive scalar curvature.
problem Proving geometric formulas and index theorems for uniformly positive scalar curvature metrics.
method Using Dirac-Schrödinger operators and relative eta invariant.
result New geometric formula for spectral flow and index formula for uniformly positive scalar curvature metrics.
Optimal volume limit found for Kähler manifolds with positive Ricci curvature.
problem Bounding the volume of Kähler manifolds with positive Ricci curvature.
method Using δ-invariants and Newton--Okounkov bodies.
result Derive the optimal volume upper bound and new characterization of the complex projective space.
It is well known that neural networks with rectified linear units (ReLU) activation functions are positively scale-invariant. Conventional algorithms like stochastic gradient descent optimize the neural networks in the vector space of weights, which is, however, not positively scale-invariant. This mismatch may lead to…
Positive sectional curvature is not preserved under the Ricci flow in dimensions seven and thirteenmath.DG
Ricci flow can change metrics with positive curvature to those without.
problem Preserving positive sectional curvature under Ricci flow in specific dimensions.
method Examined - and -invariant metrics on Aloff-Wallach spaces and Berger space.
result Found metrics with positive sectional curvature that evolve to non-positively curved metrics under Ricci flow.
Researchers prove spaces of positive scalar curvature metrics are contractible with symmetry.
problem Contractibility of spaces of invariant positive scalar curvature metrics.
method Combining equivariant Morse theory with conformal deformations and local flexibility properties.
result Spaces of invariant positive scalar curvature metrics are contractible.
Estimates Laplace eigenvalues and diameter for Lie group metrics.
problem Estimating Laplace eigenvalues and diameter for left-invariant metrics on compact Lie groups.
method Relates left-invariant metrics to positive definite matrices and uses eigenvalue properties.
result Partial answers to Eldredge's conjecture on Laplace eigenvalues and diameter.
The study establishes a link between fibered links and their concordance invariants.
problem Determining the conditions for fibered strongly quasi-positive links.
method Proved that an -component fibered link is strongly quasi-positive if and only if .
result Explicitly determined fibered prime links with at most 9 crossings and their properties.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
New mass-type invariants for cosmological space-times.
problem Characterize de Sitter solutions in space-times with a cosmological constant.
method Introduce new mass-type invariants and prove positive mass theorems.
result 1-harmonic Mass provides new characterizations and inequalities.
Let G be a discrete group, and let M be a closed spin manifold of dimension m>3 with pi_1(M)=G. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L2-rho invariant and the delocalized eta invariant associated to the Dirac operator on M in order to get information about t…
Study shows quasi-additivity of Δ-unknotting number for braids.
problem Understanding Δ-unknotting number for braids.
method Derived lower bounds on Casson invariant, showed quasi-additivity.
result Quasi-additivity of Δ-unknotting number for braids.
The paper finds bounds for a spinorial equation and applies it to a Bär-Hijazi-Lott invariant.
problem Spinorial Yamabe-type equations and their solutions.
method Analyzes spinorial Yamabe-type equations and finds positive lower bounds for λ.
result Explicit lower bounds for the Bär-Hijazi-Lott invariant are derived.
A universal collection of 4 invariants improves neural network accuracy for molecular dynamics.
problem Improving accuracy of neural networks in molecular dynamics.
method Developed a universal collection of 4 smooth scalar invariants on M(3) x M(3) and evaluated their effectiveness in a PONITA neural network architecture.
result Using a universal collection of invariants significantly improves neural network accuracy.