arXiv research

Method for generating new curves from plane curves on cylinders.

Curve shortening in metric-affine plane shrinks convex curves to points.

The study examines Bertrand Legendre curves in the unit tangent bundle over Euclidean plane.

New combinatorial type helps distinguish plane curve topologies.

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plan…

We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski $3$-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the correspon…

Paper defines curvature equivalence for Legendre curves in a plane.

A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of $τ$-functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.

Study local and global aspects of complex plane curve embeddings.

The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.

The paper extends curve deformation methods in Minkowski plane.

Study on closed $p$-elastic curves in hyperbolic and de Sitter planes.

Study a flow preserving area of plane curves, ending in a circle.

We consider the global symplectic classification problem of plane curves. First we give the exact classification result under symplectomorphisms, for the case of generic plane curves, namely immersions with transverse self-intersections. Then the set of symplectic classes form the symplectic moduli space which we compl…

The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number, called the splitting graph. By using the splitting graph, we classify the embed…

New conditions found for hyperbolic bicycle tracks.

We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $τ$ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete …

Lecture notes on curves in complex projective plane from a topological viewpoint.

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real s…

In this paper, we study a family of curves on $S^2$ that defines a two-dimensional smooth projective plane. We use curve shortening flow to prove that any two-dimensional smooth projective plane can be smoothly deformed through a family of smooth projective planes into one which is isomorphic to the real projective pla…

The paper classifies different types of cusps on plane curves.

Study on migrating elastic flows of curves across half-planes.

The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of a real plane curve as the Hessian curve of a smooth function. The plane curves …

We construct the infinite sequence of invariants for curves in surfaces by using word theory that V. Turaev introduced. For plane closed curves, we add some extra terms, e.g. the rotation number. From these modified invariants, we get the Arnold's basic invariants and some other invariants. We also express how these in…

The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.

We survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to complex plane curves. Also included are some new computations of higher-order degrees of curves, which are invariants defined in a previou…

Study inverse curve shortening flow on hyperbolic plane, classifying solitons.

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ …

Study evolutes of curves with varying smoothness.

It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed, one to another, by a contin…

Study of closed real plane curves with hyperelliptic genus three solutions.

Study on plane curves with special connections and curvatures.

The study examines vertices in curves with singular points in the Euclidean plane.

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to t…

Given a plane curve $γ: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(γ)$ of inflections which curves $\mbox{diff}(γ)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms of $\mathbb R^2$. We show that if $γ$ is an immersed curve with $D(γ)$ double points and no othe…

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also present some upper and lower bounds for the minimal number of inflection points …

The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\…

We study the integral expression of a knot invariant obtained as the second coefficient in the perturbative expansion of Witten's Chern-Simons path integral associated with a knot. One of the integrals involved turns out to be a generalization of the classical Crofton integral on convex plane curves and it is related w…

We show that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case(the double local case). Especially we give the normal form of transversely jointed two families of plane curves with second order contact at the envelope.

The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.

Study on choosing points on cubic curves, answering some questions about their flexibility.

Study generalizes Lebesgue curves to new space-filling and fractal sets.

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth-Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introducti…

New translations defined; curve shortening flow solved in hyperbolic plane.

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of s…

Shapes can roll downhill following any curve, but often return to initial orientation after crossing multiple copies.