arXiv research

New group theory insights on knot surgery results.

Study on extended weakly symmetric spaces, classifying and providing an example.

This paper studies quandles with one non-trivial column and their properties.

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.

New findings show some hyperbolic 3-manifolds can't be geometrically bounded.

Every non-trivial knot group is fully residually perfect.

Let K be a non-trivial knot in the 3-sphere and let Y be the 3-manifold obtained by surgery on K with surgery-coefficient 1. Using tools from gauge theory and symplectic topology, it is shown that the fundamental group of Y admits a non-trivial homomorphism to the group SO(3). In particular, Y cannot be a homotopy-sphe…

Smooth surfaces in simply connected 4-manifolds yield groups with non-trivial homology.

Non-trivialization probability of arc system in 3D space

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.

Non-isomorphic groups with similar profinite completions found.

The study examines groups with a specific automorphism property using BNS-invariant.

The paper revisits a claim about a principal bundle over a contractible base and finds it non-trivial.

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representa…

The paper explores idempotents in quandle rings and their connections to quandle coverings.

Study shows $(p,q)$-cables of non-trivial knots are not thin.

Study on characteristic classes for codimension-one foliations, showing non-triviality under specific conditions.

Study on characteristic classes for foliation deformations.

Let $C$ be a smooth, convex curve on either the sphere $\mathbb{S}^{2}$, the hyperbolic plane $\mathbb{H}^{2}$ or the Euclidean plane $\mathbb{E}^{2}$, with the following property: there exists $α$, and parameterizations $x(t), y(t)$ of $C$ such that for each $t$, the angle between the chord connecting $x(t)$ to $y(t)$…

New findings on Jones polynomial for 4-strand braids.

For any n\ge 2, we give infinitely many unsplittable links of n components in the 3-sphere which admit non-trivial surgery yielding the 3-sphere again and whose components are mutually distinct hyperbolic knots. Berge and Kawauchi gave 2-component hyperbolic links with those two properties. We can also give infinitely …

Definition of the partition function of U(1) gauge theory is extended to a class of four-manifolds containing all compact spaces and certain asymptotically locally flat (ALF) ones including the multi-Taub--NUT spaces. The partition function is calculated via zeta-function regularization with special attention to its mo…

We investigate geometric properties of homogeneous parabolic geometries with generalized symmetries. We show that they can be reduced to a simpler geometric structures and interpret them explicitly. For specific types of parabolic geometries, we prove that the reductions correspond to known generalizations of symmetric…

Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spect…

NT probability measures knotting in 3D arc systems.

Study on mapping class groups of infinite type surfaces.

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.

New surfaces with special geodesic and horocycle behaviors discovered.

Surface groups stable in permutations, first non-amenable example.

Let $J(π)$ be the higher order Jacobi operator. We study algebraic curvature tensors where $J(π)J(π^{\perp})=J(π^{\perp})J(π)$. In the Riemannian setting, we give a complete characterization of such tensors; in the pseudo-Riemannian setting, partial results are available. We present non-trivial geometric examples of Ri…

We prove that every acylindrically hyperbolic group that has no non-trivial finite normal subgroup satisfies a strong ping pong property, the $P_{naive}$ property: for any finite collection of elements $h_1, \dots, h_k$, there exists another element $γ\neq 1$ such that for all $i$, $\langle h_i, γ\rangle = \langle h_i …

The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivi…

Dehn fillings of knots produce manifold groups with specific trivialization properties.

We adopt a vierbein formalism to study pseudo-Finsler spaces modeled on a pseudo-Minkowski space. We show that it is possible to obtain closed expressions for most of the geometric objects of the theory, including Berwald's curvature, Landsberg's tensor, Douglas' curvature, non-linear connection and Ricci scalar. These…

We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a …

Study of volume dynamics at market spread in Bitcoin/USD.

Study magnetic curvature on Lie groups, extending Milnor's work.

This paper classifies Kähler manifolds with specific Einstein-type properties.

As a model of market price, we introduce a new type of random walk in a moving potential which is approximated by a quadratic function with its center given by the moving average of its own trace. The properties of resulting random walks are similar to those of ordinary random walks for large time scales; however, thei…

Study on LP-Sasakian manifolds with generalized η-Ricci solitons.

This paper presents the construction of the Seiberg-Witten-Floer homology of three-manifolds with non-trivial rational homology, and some properties of the invariant of three-manifolds obtained by computing the Euler characteristic. This new version corrects a mistake and some misprints that appear in the published ver…

Study of adjoint orbits in simplest non-trivial Lie algebra case.

Characterizes invariant spinors on flag manifolds.

We study Laplace eigenvalues $λ_k$ on Kähler manifolds as functionals on the space of Kähler metrics with cohomologous Kähler forms. We introduce a natural notion of a $λ_k$-extremal Kähler metric and obtain necessary and sufficient conditions for it. A particular attention is paid to the $λ_1$-extremal properties of K…

Deep models generate geometric objects with global properties.

The study explores properties of a specific type of spacetime.

Groups with certain properties have invariant subalgebra rigidity.

We analyze the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random…