Characterizes weakly linked pairs of complete graphs in 3D space.
problem Identifying pairs of complete graphs that are weakly linked.
method Algebraic characterisation and geometric analysis of linking cycles.
result Characterization of weakly linked pairs of complete graphs.
The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.
problem Characterizing and finding examples of keen weakly reducible bridge spheres.
method Analyzing bridge spheres and their properties in terms of compressing disks and width complex.
result Infinitely many examples of keen weakly reducible bridge spheres for links in b-bridge position.
We show that if a link L with non-zero Alexander polynomial admits a locally flat cobordism to a `weakly m-split link', then the cobordism must have genus at least (m-1)/2. This generalises a recent result of J. Pardon.
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.
Standard position for surfaces extended to weakly generalized alternating links.
problem Proving standard position for surfaces in a broader class of links.
method Extending techniques from classical alternating links to weakly generalized alternating links.
result All weakly generalized alternating links are prime, and Conway spheres interact with diagrams as in classical setting.
All link types arise from semiholomorphic polynomials.
problem Proving every link type can be represented by semiholomorphic polynomials.
method Constructive proof showing every link type arises from a weakly isolated singularity of a semiholomorphic polynomial.
result Every link type in the 3-sphere arises as the link of a weakly isolated singularity of a semiholomorphic polynomial.
Paper describes links of mixed polynomials with specific properties.
problem Understanding the links of mixed polynomials with nice Newton boundaries.
method Analyzes links constructed from sequences of links associated with compact 1-faces of the Newton boundary.
result Links of singularities of inner non-degenerate mixed polynomials can be described using a specific procedure.
New invariant measures doubly slice links, disproving previous bounds.
problem Understanding doubly slice links and their invariants.
method Introduced new invariant gst to measure doubly slice links and disproved previous bounds. result Examples of links with large doubly slice genus but gst=1. New examples show no upper bounds on link volumes on incompressible surfaces.
problem Finding upper bounds on volumes of links on incompressible surfaces.
method Examined weakly generalised alternating and fully augmented links on incompressible surfaces.
result Found infinite families of links on incompressible surfaces with no upper bounds on volume.
New method detects essential tori in mixed singularity links.
problem Detecting essential tori in mixed singularity link complements.
method Analyzing properties of defining mixed polynomials.
result Explicit criteria for essential tori existence.
A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A-B slice. A stronger version of the conjecture, that the Borromean r…
We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometr…
The paper develops a theory of skein adequate links in thickened surfaces and proves Tait conjectures.
problem Establishing Tait conjectures for adequate links in thickened surfaces.
method Applying Kauffman bracket skein algebras to develop a theory of skein adequate links and proving Tait conjectures.
result The crossing number is additive under connected sum for adequate links in thickened surfaces.
Modified proof for a broader class of links.
problem Recovering twist number and volume bounds.
method Modified proof for a broader class of links.
result Modified proof works for a broader class of links.
Proves weakly non-collapsed RCD spaces are strongly non-collapsed.
problem Proving the equivalence of weakly non-collapsed and strongly non-collapsed RCD spaces.
method Analyzes properties of RCD spaces and uses auxiliary results.
result Confirms conjecture about RCD spaces being strongly non-collapsed.
The paper examines linking numbers in grid models and finds polynomial moments.
problem Analyzing linking numbers in grid models.
method Examined linking numbers as a random variable on isotopy classes of 2-component links, computed moments and limits.
result The uth moment of the linking number is a polynomial in the grid size with degree d≤u, and all odd moments vanish. The goal of this paper is to describe and clarify as much as possible the 3-dimensional topology underlying the Helmholtz cuts method, which occurs in a wide theoretic and applied literature about Electromagnetism, Fluid dynamics and Elasticity on domains of the ordinary space. We consider two classes of bounded domain…
Study shows bounds on volumes of weakly generalised alternating knots.
problem Volume bounds for weakly generalised alternating knots.
method Analysis of weakly generalised alternating knots in 3-manifolds.
result Upper volume bound does not hold for weakly generalised alternating knots.
The study determines conditions for hyperbolicity of links in thickened surfaces with boundary.
problem Conditions for hyperbolicity of links in thickened surfaces with boundary.
method Simple conditions and embeddings of surfaces in ambient manifolds.
result Conditions guaranteeing hyperbolicity of links in thickened surfaces and fiber bundles.
Suppose a link K in a 3-manifold M is in bridge position with respect to two different bridge surfaces P and Q, both of which are c-weakly incompressible in the complement of K. Then either P and Q can be properly isotoped to intersect in a nonempty collection of curves that are essential (including non-meridional) on …
The paper shows some Montesinos links can't be doubly sliced strongly.
problem Understanding double sliceness for Montesinos links.
method Using branched double covers and Seifert fibered spaces.
result A large family of Montesinos links are not strongly doubly slice.
Let L be a link in the 3-sphere that is in thin position but not in bridge position and let P be a thin level sphere. We generalize a result of Wu by giving a bound on the number of disjoint irreducible compressing disks that P can have, including identifying thin spheres with unique compressing disks. We also give con…
The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number…
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the …
Study on scalar curvature decay on non-compact manifolds linked at infinity.
problem Understanding scalar curvature decay on non-compact manifolds with topological linking at infinity.
method Analyzing polynomial decay, developing obstruction theory, using μ--bubble exhaustions, and index theory. result Topological linking at infinity forces polynomial decay of scalar curvature on manifolds of weakly bounded geometry.
The dynamic network of relationships among corporations underlies cascading economic failures including the current economic crisis, and can be inferred from correlations in market value fluctuations. We analyze the time dependence of the network of correlations to reveal the changing relationships among the financial,…
Researchers debunk a generalized Property R conjecture for 2-component links.
problem The generalized Property R conjecture for 2-component links in S3. method Disproved a further generalization of the GPRC using specific counterexamples.
result 2-component framed links in S3 surging to a connected sum of homology S1imesS2's are not handleslide equivalent to a split link. We show that every p-fold strictly-cyclic branched covering of a b-bridge link in S3 admits a p-symmetric Heegaard splitting - in the sense of Birman and Hilden - of genus g=(b−1)(p−1). This gives a complete converse of one of the results of the two authors. Moreover, we introduce the concept of weakly p-symmetric…
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details)…
Weakly Einstein Kähler surfaces are characterized and classified.
problem Characterizing and classifying weakly Einstein Kähler surfaces.
method Several conditions and constructions to characterize and classify weakly Einstein Kähler surfaces.
result Classification of weakly Einstein Kähler surfaces with specific properties and construction of new examples.
The study examines weakly Einstein Lie groups and proves non-existence for certain types.
problem Characterizing and proving the non-existence of weakly Einstein Lie groups.
method Analyzing left-invariant metrics on Lie groups and using algebraic properties.
result No weakly Einstein non-abelian 2-step nilpotent Lie groups exist.
Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.
problem Characterizing submanifolds in space forms with certain geometric properties.
method Classification based on Chen's equality and semisymmetric conditions.
result Classification of weakly Einstein submanifolds in space forms.
The study explores weakly p-Kähler hyperbolic manifolds.
problem Generalization and application of weakly p-Kähler hyperbolic manifolds. method Investigation of generalizations and applications.
result Exploration of weakly p-Kähler hyperbolic manifolds. Every time drivers take to the road, and with each mile that they drive, exposes themselves and others to the risk of an accident. Insurance premiums are only weakly linked to mileage, however, and have lump-sum characteristics largely. The result is too much driving, and too many accidents. In this paper, we introduce…
This paper studies the topological properties of the World Trade Web (WTW) and its evolution over time by employing a weighted network analysis. We show that the WTW, viewed as a weighted network, displays statistical features that are very different from those obtained by using a traditional binary-network approach. I…
Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.
problem Characterizing weakly weighted Einstein-Finsler metrics.
method Showed isotropic S-curvature under certain conditions. Characterized via navigation expressions and α and β. result Weakly weighted Einstein-Kropina metrics have isotropic S-curvature and can be completely characterized.
The paper examines node2vec embeddings for community detection in networks.
problem Theoretical understanding of node2vec embeddings for community detection.
method Analysis of node2vec embeddings for community recovery in stochastic block models.
result k-means clustering on node2vec embeddings gives weakly consistent community recovery for stochastic block models.
Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
Study on extended weakly symmetric spaces, classifying and providing an example.
problem Understanding geometric properties of extended weakly symmetric spaces.
method Classification and presentation of a non-trivial example.
result Existence of extended weakly symmetric spaces established.
There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit…
Classifies weakly Einstein hypersurfaces in spaces of constant curvature.
problem Identifying weakly Einstein hypersurfaces in spaces of constant curvature.
method Complete classification through tensor analysis and geometric properties.
result Hypersurfaces are either products of spaces of constant curvature or rotation hypersurfaces.
We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of s-representations.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. New examples of weakly Einstein conformal products are constructed.
problem Classifying weakly Einstein conformal products.
method Established classification results and constructed new examples.
result No proper weakly Einstein manifolds have harmonic curvature.
In this article we study almost contact manifolds admitting weakly Einstein metrics. We first prove that if a (2n+1)-dimensional Sasakian manifold admits a weakly Einstein metric then its scalar curvature s satisfies −6⩽s⩽6 for n=1 and $-2n(2n+1)\frac{4n^2-4n+3}{4n^2-4n-1}\leqslant s \leqslant …
The object of the present note is to discuss about the defining condition of weakly cyclic Ricci symmetric manifolds and weakly cyclic Z-symmetric manifolds \cite{DMS15} and the existence of such notion by proper examples.