arXiv research

Quaternionic curves with specific torsion properties don't exist.

Let ${\rm SL(2, \mathbb H)}$ be the group of $2 \times 2$ quaternionic matrices with Dieudonné determinant $1$. The group ${\rm SL(2, \mathbb H)}$ acts on the five dimensional hyperbolic space by isometries. We investigate extremality of Jørgensen type inequalities in ${\rm SL(2, \mathbb H)}$. Along the way, we derive …

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of the five dimensional (real) hyperbolic space. We obtain discreteness criteria f…

Study on Bertrand lightcone framed curves in Lorentz-Minkowski 3-space.

Study of generalized Bishop frames on curves in 4D space.

Study on focal surfaces of lightcone framed surfaces in Lorentz-Minkowski 3-space.

Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…

Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.

A. Henrich proved the existence of the universal finite-type invariant of order one for virtual knots. We extend the construction and the methods of her paper to framed virtual knots. To do so, we introduce the notions of virtual strings and based matrices for framed flat virtual knots.

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…

Study of generalized Bishop frames on time-like curves in 4D Lorentz space.

In this paper, we have first given easily the characterization of special curves with the help of the Rotation minimizing frame (RMF). Also, rectifying-type curves are generalized n-dimensional space $R_{n}$.

We generalize three invariants, first discovered by A. Henrich, to the long and/or framed virtual knot case. These invariants are all finite-type invariants of order one, and include a universal one. The generalization will require us to extend the notion of a based matrix of a virtual string, first introduced by V. Tu…

In this study, we consider AW(k)-type curves according to the Bishop Frame in Euclidean space E^3. We give the relations between the Bishop curvatures k_1, k_2 of a curve in E^3.

In this study we consider AW(k)-type curves according to parallel transport frame in Euclidean space E^4. We give the relations between the parallel transport curvatures of these kinds of curves.

The paper explores Bertrand and framed curves in 3D space.

The paper parametrizes spaces of maximal framed representations for a specific type of surface group.

In this paper we introduce two theories of finite type invariants for framed links with fixed linking matrix. We show that these thepries are related to the theory of Vassiliev invariants of framed links. We also study the corresponding spaces of ``chord diagrams''.

Modified knotoids with framing and coframing for quantum invariants.

CNN improves frame selection for ultrasound elastography.

Study on singular points of translation surfaces under linearly dependent conditions.

New invariant for 4-manifolds with framed links, stronger than existing invariants.

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-secti…

In this work, we move beyond the traditional complex-valued representations, introducing more expressive hypercomplex representations to model entities and relations for knowledge graph embeddings. More specifically, quaternion embeddings, hypercomplex-valued embeddings with three imaginary components, are utilized to …

The paper connects hyperbolic spinors to non-null framed curves in Minkowski 3-space.

We construct a compactification $M^{μss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $γ\colon M^{ss} \to M^{μss}$, where $M^{ss}$ is the moduli space of S-equiva…

We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.

We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The proc…

The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As first …

Satellite formula connects knot concordance invariants to surgery.

For all left-invariant Riemannian metrics on three-dimensional unimodular Lie groups, there exist particular left-invariant orthonormal frames, so-called Milnor frames. In this paper, for any left-invariant Riemannian metrics on any Lie groups, we give a procedure to obtain an analogous of Milnor frames, in the sense t…

A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed l…

The paper explores Parseval frames on vector bundles, proving their existence for certain cases.

It is well-known that a knot in a contact manifold $(M,C)$ transverse to a trivialized contact structure possesses the natural framing given by the first of the trivialization vectors along the knot. If the Euler class $e_C\in H^2(M)$ of $C$ is nonzero, then $C$ is nontrvivializable and the natural framing of transvers…

We consider the orthonormal frame bundle F(M) of a Riemannian manifold M. A construction of Sasaki defines a canonical Riemannian metric on F(M). We prove that for two closed Riemannian n-manifolds M and N, the frame bundles F(M) and F(N) are isometric if and only if M and N are isometric, except possibly in dimensions…

Harmonic maps link Teichmüller spaces to framed representations.

We consider axisymmetric stationary dirty black holes with regular non-extremal or extremal horizons, and compute their on-horizon Petrov types. The Petrov type (PT) in the frame of the observer crossing the horizon can be different from that formally obtained in the usual (but singular in the horizon limit) frame of a…

We show that for a large class of contact 3-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's $J^+$-type invariants of wave fronts on a surface $F$ is isomorphic to the group of Vassiliev invariant…

The paper defines evolutes and involutes for framed curves and their properties.

Researchers prove a method to upgrade Morse-Bott homology to stable homotopy invariants.

In the bordered Floer theory, gluing thickened torus of positive meridional Dehn twist to the boundary of a knot complement result in the knot complement of increased framing. For a fixed knot K, we construct a direct system of positively framed knot complements and study the direct limit. We also study the morphism sp…

Researchers find a Steenrod square for link Floer homology.

The paper explores noncommutative geometry of frame bundles using C*-algebras.

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map of Weingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundament…

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define…

A 3D space of hyperbolic manifolds is connected but not path-connected.

In this paper, we define a new type of ruled surface called ruled surface by using the alternative frame of a base curve. Then, we study its differential geometric properties such as striction line, distribution parameter, fundamental forms, Gaussian and mean curvatures. Moreover, we find geodesic curvatures, normal cu…

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were in…