When the signed weighted resolution set was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al, we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achi…
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48 results for “sign of invariant”
Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.
problem Understanding the relationship between Gaussian curvature and singularities of Gauss maps of cuspidal edges.
method Analyzes geometric invariants and types of singularities of Gauss maps to define and characterize positivity/negativity of cusps.
result Defines and characterizes positivity/negativity of cusps of Gauss maps by geometric invariants of cuspidal edges, and shows relation between sign of cusps and Gaussian curvature.
Signed Heights of Knotoidsmath.GT
Signed heights of knotoids are defined and studied.
problem Understanding the signed height of knotoids.
method Defined positive and negative parts of height, proved they determine unsigned height, provided lower bounds with polynomials, studied associated sequences.
result Positive and negative parts of height determine unsigned height.
Sign equivariant networks improve model expressiveness for spectral geometric learning.
problem Limited expressiveness of sign invariant models for tasks like graph link prediction.
method Developed sign equivariant neural network architectures based on new analytic sign equivariant polynomials.
result Sign equivariant models achieve theoretical benefits in spectral geometric learning tasks.
Researchers analyze sign choices for O-planes in orientifolds, proving stabilisation and deriving topological constraints.
problem Assigning sign choices to O-planes in orientifolds of type II string theory.
method Investigating sequences of invariant p-gerbes and coboundary maps to derive sign choices and topological constraints.
result Sign choice homomorphisms stabilise with the dimension of the orientifold and topological constraints on sign configurations are derived.
The paper extends knot algebra with signs for more complex computations.
problem Computing knot invariants with additional algebraic structure.
method Generalizes bordered knot algebra with signs and integral coefficients.
result Effective computations of the new invariant are described.
Given an affine isometry of with hyperbolic linear part, its Margulis invariant measures signed Lorentzian displacement along an invariant spacelike line. In order for a group generated by hyperbolic isometries to act properly on , the sign of the Margulis invariant must be constant over the group. We show…
We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…
New invariant measures loop iterations in algebraic structures.
problem Measuring the asymptotic behavior of loop iterations in algebraic structures.
method Introduced sign stability and cluster stretch factor to measure loops.
result Cluster algebraic entropies match cluster stretch factor.
When solving data analysis problems it is important to integrate prior knowledge and/or structural invariances. This paper contributes by a novel framework for incorporating algebraic invariance structure into kernels. In particular, we show that algebraic properties such as sign symmetries in data, phase independence,…
New invariant counts SU(2) representations for links.
problem Counting irreducible SU(2) representations for links.
method Signed count of irreducible representations with fixed traces.
result Invariant is a sum of multivariable signatures for specific links.
Extends interior polynomial to signed bipartite graphs and connects to HOMFLY polynomial.
problem Invariants of signed bipartite graphs and their relation to HOMFLY polynomial.
method Extending interior polynomial to signed bipartite graphs and showing equality to HOMFLY polynomial part.
result Interior polynomial of signed bipartite graphs equals part of HOMFLY polynomial for planar case.
New neural network architectures use signed permutation representations for finite groups, improving performance.
problem Designing and optimizing neural networks for finite groups with signed permutation representations.
method Introduces -invariant deep neural networks with densely connected layers and signed permutation representations.
result Signed permutation representations lead to significantly better performance in classification tasks.
Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alterna…
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
problem Learning invariant graph representations from eigenvectors.
method SignNet and BasisNet neural architectures that are invariant to sign flips and basis symmetries.
result Proven to be universal, approximating any continuous function of eigenvectors with desired invariances.
We associate to any Riemannian symmetric space (of finite or infinite dimension) a L-algebra, under the assumption that the curvature operator has a fixed sign. L-algebras are Lie algebras with a pleasant Hilbert space structure. The L-algebra that we construct is a complete local isomorphism invariant and …
A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by coun…
New invariant for spin 3-manifolds using super 3-cocycles.
problem Constructing an invariant for spin 3-manifolds.
method State sum construction based on super 3-cocycles and combinatorial representation.
result The invariant is sensitive to the spin structure.
New nodal domain theorems for symmetric matrices via signed graphs.
problem Establish nodal domain theorems for symmetric matrices.
method Explore signed graph structure to define nodal domains for any function.
result Improved lower bound estimates for the number of strong nodal domains.
Quandles and Linking Numbermath.GT
We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.
New signs and gradings enable detailed comparison in Heegaard Floer theory.
problem Comparing decategorified Heegaard Floer theory with modern TQFTs.
method Added signs and gradings to interval gluing theorem over Z.
result Detailed comparison possible with modern TQFTs.
Computed Casson-Lin invariant for Hopf link.
problem Computing the Casson-Lin invariant for specific links.
method Direct computation and analysis of the Hopf link.
result Determined the sign in the linking number formula.
In a recent joint work with V. Turaev (cf. math.DG/9810114) we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion it has the advantage of having a well-defined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector …
Study finds infinite sign-changing solutions for a specific equation on manifolds.
problem Existence of sign-changing solutions for a Yamabe-type equation on manifolds.
method Analyzes a specific Yamabe-type equation on manifolds with proper isoparametric functions and positive focal submanifolds.
result Proves the existence of infinite sign-changing solutions for the equation when .
This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesi…
Paper finds infinitely many solutions changing sign for critical fractional equations.
problem Critical fractional equations with sign-changing solutions.
method Reduction to equivalent problem on sphere, blow-up arguments, Pohozaev's identity, regularity results, symmetries of sphere.
result Unbounded sequence of sign-changing solutions for critical problems.
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fix…
New finite symmetries found in 4-manifolds.
problem Understanding symmetries in 4-dimensional spaces.
method Analyzing connected sums and finite group actions.
result Found infinitely many non-equivalent smooth structures and actions.
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…
We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking number of the link. Our construction generalizes that of X.-S. Lin who defined a sim…
Yamabe Classification and Prescribed Scalar Curvature in the Asymptotically Euclidean Settingmath.DG
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yam…
Researchers solve the negative Yamabe case for scalar curvature prescription.
problem Prescribing scalar curvature on manifolds with negative Yamabe invariant.
method A new variational approach to the problem.
result Existence of solutions for sign-changing scalar curvature functions, but not uniqueness.
We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don't cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you con…
In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invar…
Results are obtained on extending flat vector bundles or equivalently general representations from the fundamental group of S, a connected subsurface of the connected boundary of a compact, connected, oriented 3-dimensional manifold, to the whole manifold M. These are applied to representations of fundamental groups of…
Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
Singular Yamabe problems involve changing sign solutions with interesting geometric properties.
problem Solving Yamabe problems with changing sign solutions and their geometric implications.
method Analyzing the behavior of conformal factors and zero loci in various dimensions.
result Zero loci of solutions are critical for conformal functionals and can be Willmore energy minimizers.
We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits.…
In a conformal class of metrics with positive Yamabe invariant, we derive a necessary and sufficient condition for the existence of metrics with positive Q curvature. The condition is conformally invariant. We also prove some inequalities between the Green's functions of the conformal Laplacian operator and the Paneitz…
Introduces a metric on vector-valued one-forms for functional data analysis.
problem Metric on vector-valued one-forms for functional data analysis.
method Diffeomorphism-invariant Riemannian metric calculation and geodesic equations.
result Geodesically and metrically incomplete space with specific curvature properties.
The paper classifies special Randers metrics on Lie groups.
problem Classifying Randers metrics of Douglas type on Lie groups.
method Classification through invariant hyper-Hermitian metrics.
result Formulas for flag curvature and same sign curvature in some directions.
Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, incl…
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
Study finds least-energy nodal solutions to the Yamabe problem on manifolds with boundary.
problem Existence of sign-changing solutions to the Yamabe problem on manifolds with boundary.
method Variational approach, analysis of conformal invariants, and sharp energy estimates.
result Existence of least-energy nodal solutions when the manifold is positive and the boundary has non-negative constant mean curvature.
Shephard groups are unitary reflection groups arising as the symmetries of regular complex polytopes. For a Shephard group, we identify the representation carried by the principal ideal in the coinvariant algebra generated by the image of the product of all linear forms defining reflecting hyperplanes. This representat…
The abstract provides a method to represent knots and generalizes knot invariants.
problem Representing knots and generalizing knot invariants.
method Signed chord diagrams and moves to connect diagrams for the same knot.
result Every knot can be represented by signed chord diagrams and connected by moves.
Paper introduces stable vectorization for multiparameter PH using signed barcodes.
problem Lack of stable vectorization methods for multiparameter persistent homology.
method Signed barcodes as measures for stable vectorization of MPH.
result Stable feature vectors from signed barcodes improve performance in data science.
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of t…