arXiv research

Bertrand framed surfaces defined in Euclidean 3-space with applications.

New proof of Lie-Tresse theorem with computational advantages.

By combining the ideas of Cartan's equivalence method and the method of the equivariant moving frame for pseudo-groups, we develop an efficient method for solving equivalence problems arising from horizontal Lie pseudo-group actions. The key is a pseudo-group analog of the classic result that characterizes congruence o…

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

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

The paper develops a new Floer theory for 3-manifolds with involutions.

Proves convergence of normal forms for infinite-dimensional Lie pseudo-group actions.

Study of equivariant movie moves for involutive links.

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

Defines non-parabolic curves in spatial hybrid space with applications.

Discussing moving frames for curve and surface invariants.

New quantum invariant for framed 3-manifolds using ideal triangulations.

New framed moves extend classical knot theory results.

In this paper, we investigate some characterizations of involute -- evolute curves in dual space. Then the relationships between dual frenet frame and darboux vectors of these curves are found.

Specialized knot theory theorems for strongly involutive links.

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…

Analyses cohomology relations for moving frames and coframes.

Minimal sets of moves for rotational Reidemeister diagrams are identified.

New formula connects Loewner energy to moving frames' renormalised energy.

New method studies moving points on curves using rotating frames.

Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…

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…

In this paper, we take into account the opinion of involute-evolute curves which lie on fully surfaces and by taking into account the Darboux frames of them we illustrate these curves as special involute-evolute partner D-curves in E3. Besides, we find the relations between the normal curvatures, the geodesic curvature…

Frames for $\R^n$ can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tang…

Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the…

The paper uses Cartan moving frames to analyze data manifolds and neural network outputs.

New neural network processes 3D volumes with improved equivariance.

The aim of this paper is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. Involute trajectory ruled surfaces generated by the Frenet trihedron, moving along spacelike involutes of a given timelike space curve, is stated according to Lorentzian timelike angle …

We introduce a new knot diagram invariant called the Self-Crossing Index (SCI). Using SCI, we provide bounds for unknotting two families of framed unknots. For one of these families, unknotting using framed Reidemeister moves is significantly harder than unknotting using regular Reidemeister moves. We also investigate …

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…

In our previous work, we have defined a nonlinear connection of Finsler manifold which preserves the Finsler metric $L=L(x,dx)$. To make the method easier and more useful in applications, moving frame (vielbein) $θ^a={e^a}_μdx^μ$ formalism for the nonlinear connection is newly considered. We derive formulae to calculat…

This paper is devoted to apply the equivariant moving frame method to study the local equivalence problem of third order ordinarily differential equation under the pseudo-group of fiber preserving transformations.

The paper studies Riemannian four-manifolds and their twistor spaces using a moving frame approach.

Let $Ω\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial frame on $Ω$, that is a set of $m$ linearly independent vector fields prescribed on $Ω$ ($m\leq n$). We consider the issue of describing the set of all maps $F:Ω\to\mathbb{R}^n$ with the property that each of the given vector fields is an eigenvecto…

New method for curve comparison using iterated integrals and moving frames.

This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an equivariant map from the space of submanifolds to the group itself and introduce two algor…

Study curvature of piecewise metrics using moving frames.

Study geometry of surfaces glued along a curve.

Normal forms and invariants for nondegenerate hypersurfaces in C^2.

Diagonalizes metrics of 3D Lorentzian manifolds.

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…

Generators and relations found for foam categories.

The Frenet frame generalizes the Park transform for multi-phase circuits.

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 …

A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.

We classify 3-braids up to (2,2)-move equivalence and in particular we show how to adjust the Harikae-Nakanishi-Uchida conjecture so it holds for closed 3-braids. As important steps to classify 3-braids up to (2,2)-move equivalence we prove the conjecture for 2-algebraic links and classify (2,2)-equivalence classes for…

Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.

This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution on the knot Floer homology of $K$ which corresponds to moving the basepoints by o…