Related papers: Arnold-type Invariants of Curves on Surfaces
Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants…
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a…
This paper introduces the $J^{2+}$-invariant for oriented pairs of generic immersions. This invariant behaves like Arnold's $J^+$-invariant for generic immersions as it is invariant when going through inverse tangencies and triple points,…
We explore Arnold's $J^+$-invariant of immersions -- planar smooth closed curves with non-vanishing derivative, at most double points and only transverse intersections -- and computation methods like Viro's sum, among others. Only basic…
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…
By integrating curvatures multiplied non-trivial densities, we introduce an integral expression of the Arnold strangeness that is a celebrated plane curve invariant. The key is a partition function by Shumakovitch to reformulate Arnold…
It is known that for any smooth sphere eversion, the number of quadruple point jumps is always odd. In this paper, we define an integer-valued function that detects and classifies jumps involving quadruple points and triple-line tangencies.…
In their paper "Integrating curvature: From Umlaufsatz to J+ invariant" Lanzat and Polyak introduced a polynomial invariant of generic curves in the plane as a quantization of Hopf's Umlaufsatz, and showed that Arnold's J+ invariant could…
We give a classification of generic coadjoint orbits for the group of area-preserving diffeomorphisms of a closed non-orientable surface. This completes V. Arnold's program of studying invariants of incompressible fluids in 2D. As an…
We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many…
We define a new finite type invariant for stably homeomorphic class of curves on compact oriented surfaces without boundaries and extend to a regular homotopy invariant for spherical curves.
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…
This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit…
We introduce a framework on dual complexes for studying Arnold-type invariants of immersed curves and immersed surfaces via local finite-difference structures associated with Alexander numberings. For generic immersed plane curves and…
We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and…
We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
We explore how the invariants $J^+$, $J^-$, $\mathcal{J}_1$ and $\mathcal{J}_2$ of immersions -- generic (at most double points and only transverse intersections) planar smooth closed curves with non-vanishing derivative -- change under…
This paper demonstrates an approach for developing a framework to produce invariants of base-point-free generic spherical curves under some chosen local moves from Reidemeister moves using based chord diagrams. Our invariants not only…
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational…