arXiv research

Local index formula for Lorentzian Dirac operators on spacetimes.

No nontrivial automorphisms for cubic surfaces moduli space.

Study on warped product Yamabe solitons with constant fiber curvature.

We consider the following question: when is the manifold obtained by gluing together two knot complements an $L$-space? Hedden and Levine proved that splicing 0-framed complements of nontrivial knots never produces an $L$-space. We extend this result to allow for arbitrary integer framings. We find that splicing two in…

We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off-diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild me…

In this article, it is proved that the eigenvalue variety of the exterior of a nontrivial, non-Hopf, Brunnian link in $\mathbb{S}^{3}$ contains a nontrivial component of maximal dimension. This generalises, for Brunnian links, the nontriviality of the $A$-polynomial of a nontrivial knot in $\mathbb{S}^{3}$.

Lower bounds on cone density for nontrivial complements in low dimensions.

New nontrivial breathers found for Ricci flow on noncompact manifolds.

Smooth HP^2 bundle over S^4 with nontrivial A-genus found.

In this paper, we discuss the general existence theory of Dirac-harmonic maps from closed surfaces via the heat flow for $α$-Dirac-harmonic maps and blow-up analysis. More precisely, given any initial map along which the Dirac operator has nontrivial minimal kernel, we first prove the short time existence of the heat f…

It was asked by J.Birman, Williams, and L.Rudolph whether nontrivial Lorentz knots have always positive signature. Lorentz knots are examples of positive braids (in our convention they have all crossings negative so they are negative links). It was shown by L.Rudolph that positive braids have positive signature (if the…

The real cohomology of the space of imbeddings of S^1 into R^n, n>3, is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomol…

Echo state networks are computationally lightweight reservoir models inspired by the random projections observed in cortical circuitry. As interest in reservoir computing has grown, networks have become deeper and more intricate. While these networks are increasingly applied to nontrivial forecasting tasks, there is a …

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

We determine when certain state cycles represent nontrivial Khovanov homology classes by analyzing features of the state graph. Using this method, we are able to produce hyperbolic knots with arbitrarily many diagonals containing nontrivial state cycle homology classes. This gives lower bounds on the Khovanov width of …

Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in $S^{3}$ is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-po…

Study shows nontrivial intersections of subgroups on homogeneous spaces.

A metric projective structure is a manifold equipped with the unparametrised geodesics of some pseudo-Riemannian metric. We make acomprehensive treatment of such structures in the case that there is a projective Weyl curvature nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invaria…

This note gives a proof that the $A$-polynomial of any nontrivial knot in $S^3$ has nontrivial $M$-degree.

We give examples of knots in a genus 2 handlebody which have nontrivial Dehn surgeries yielding handlebodies and show that these knots are not 1--bridge.

New method identifies latent components in PNL mixtures without strong assumptions.

We construct nontrivial cohomology classes of the space $Imb(S^1,\R^n)$ of imbeddings of the circle into $\R^n$, by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number $n\geq 4$, a de Rham cohomology class on $Imb(S^1,\R^n)…

A new single-pass algorithm improves sparse PCA under limited computational resources.

The study proves conditions for nontrivial solutions on Riemannian manifolds.

In this paper, we prove that $w(K) =4w(J)$, where $w(.)$ is the width of a knot and $K$ is the Whitehead double of a nontrivial knot $J$.

The term "special biconformal change" refers, basically, to the situation where a given nontrivial real-holomorphic vector field on a complex manifold is a gradient relative to two Kähler metrics, and, simultaneously, an eigenvector of one of the metrics treated, with the aid of the other, as an endomorphism of the tan…

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [1],[2]. Building on this and using the chirality of torus knots and links [3],[4], we prove that nontrivial embeddings of simple 3-…

Theta graph diffeomorphism shows nontrivial mapping class of 4-sphere.

In this paper we present a sequence of link invariants, defined from twisted Alexander polynomials, and discuss their effectiveness in distinguish knots. In particular, we recast and extend by geometric means a recent result of Silver and Williams on the nontriviality of twisted Alexander polynomials for nontrivial kno…

We show that there are infinitely many nonisomorphic quandle structures on any topogical space $X$ of positive dimension. In particular, we disprove the conjecture, asserting that there are no nontrivial quandle structures on the closed unit interval $[0,1]$.

New topological biquandles created using skew braces.

The paper studies invariant convex sets in representations with nontrivial copolarity.

Classification of Finslerian spaces with nontrivial concircular transformations.

We characterize Willmore tori in the 4-sphere with nontrivial normal bundle as Twistor projections of elliptic curves in complex projective space or as inverted minimal tori (with planar ends) in Euclidean 4-space.

We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we prove a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

We explore under what conditions one can obtain a nontrivial knot, given a collection of $n$ vectors. First, we show how to get a crossing from any 3 vectors equal in magnitude, by arbitrarily picking 2 vectors and identifying the sufficient and necessary criteria for picking a third vector that will guarantee a crossi…

Let $G$ be a compact connected semisimple Lie group with Lie algebra $\mathfrak{g}$. Let $\mathcal{O}\subset\mathfrak{g}^*$ be a coadjoint orbit. The action of $G$ on $\mathcal{O}$ induces a morphism $ρ:G\to \mathrm{Homeo}(\mathcal{O})$. We prove that the induced map $π_1(ρ):π_1(G)\toπ_1(\mathrm{Homeo}(\mathcal{O}))$ i…

Origami can create complex knots, with minimum creases defining a new knot invariant.

We prove that a nontrivial complete generalized quasi Yamabe gradient soliton (M; g) must be a quasi Yamabe gradient soliton on each connected component of M and that a nontrivial complete locally conformally at generalized quasi Yamabe gradient soliton has a special warped product structure.

New domains found in hyperbolic space solve a specific elliptic problem.

Study finds nontrivial $n$-harmonic maps from $S^n$ to closed manifolds.

The paper introduces a new filtration for knot invariants and proves the existence of nontrivial knots.

Found a new connected component in symplectic structures.

We consider symplectic Floer homology in the lowest nontrivial dimension, that is to say, for area-preserving diffeomorphisms of surfaces. Particular attention is paid to the quantum cap product; we show that it distinguishes the trivial element of the mapping class group from any nontrivial one.

Persistent elements are ubiquitous in knot groups, especially for hyperbolic knots.

Study YB operators and their deformations, finding integrable and nontrivial cases.

The paper classifies 3D complete gradient Yamabe solitons.