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…
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Paper develops invariants for spherical curves using chord diagrams.
problem Developing invariants for spherical curves under local moves.
method Using based chord diagrams and local moves from Reidemeister moves.
result Invariants include both classical and new spherical curve invariants.
Machine learning predicts arithmetic curve invariants with high accuracy.
problem Classifying arithmetic curves based on their invariants.
method Training machine learning algorithms on datasets of elliptic and genus 2 curves.
result High accuracy in classifying curves, including rank, torsion, and integral points.
The delta invariant of curves on rational surfaces II: Poincaré series and topological aspectsmath.GT
Study delta invariant of curves on rational surfaces using topological methods.
problem Calculate delta invariant for curves embedded in rational singularities.
method Use topological techniques and Poincaré series.
result Develop formulae for delta invariant in terms of embedded data.
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 higher-order generalisation of a part of Arnold's invariants that are first-ord…
Formula connects surface and curve invariants via slice transitions.
problem Computing surface invariants from curve invariants.
method Introducing differential measures for local changes across singular slice transitions.
result Explicit formula for surface invariant change during quadruple-point events.
Flow on curves in inversive geometry converges to loxodromics.
problem Gradient flow for curve length in inversive geometry.
method Invariant gradient flow for invariant length functional.
result Solutions exist for all time and converge to loxodromic curves.
We give a complete description of all order 1 invariants of spherical curves. We also identify the subspaces of all J-invariants and S-invariants, and present two equalities satisfied by any spherical curve.
Study on algebraic curves' invariants and vanishing criteria.
problem Vanishing criteria for Griffiths infinitesimal invariants of algebraic curves.
method Analysis of moduli space of smooth genus 4 curves, study of normal functions.
result Vanishing criteria for the Griffiths infinitesimal invariants of Ceresa normal function.
Pedal curves derived from ellipses are invariant in area.
problem Finding invariant areas of pedal curves derived from ellipses.
method Analytical proof and explicit area expressions.
result Pedal curves derived from ellipses are invariant in area.
We provide techniques for studying the nonnegatively curved left-invariant metrics on a compact Lie group. For "straight" paths of left-invariant metrics starting at bi-invariant metrics and ending at nonnegatively curved metrics, we deduce a nonnegativity property of the initial derivative of curvature. We apply this …
Boundedness of geometric invariants near a singularity which is a suspension of a singular curvemath.DG
The paper examines geometric invariants near a specific type of singular point.
problem The behavior of geometric invariants near a singular point of a surface or curve.
method Analysis of geometric invariants for surfaces and curves that are suspensions of singular curves.
result Evaluation of the orders of Gaussian and mean curvatures for the studied surfaces and curves.
New combinatorial type helps distinguish plane curve topologies.
problem Distinguishing the topology of plane curves.
method Introducing G-combinatorial type using modified plumbing graphs.
result Invariant of G-combinatorial type under certain homeomorphisms.
The paper is devoted to differential geometric invariants determining a Frenet curve in up to a direct similarity These invariants can be presented by the Euclidean curvatures in terms of an arc lengths of the spherical indicatrices. Then, these invariants expressed by focal curvatures of the curve. And then, we give t…
Study structural invariants of Goursat distributions related to curve singularities.
problem Understanding local invariants of Goursat distributions.
method Investigate structural invariants akin to curve singularities on surfaces.
result Relate structural invariants to small growth invariants in the sequel.
This paper investigates the equivalence between Yamada polynomial and Jones polynomial of associated links for brunnian θ-curves.
problem Understanding the relationship between Yamada polynomial and Jones polynomial for θ-curves.
method Investigates the equivalence between the normalized Yamada polynomial of θ-curves and the Jones polynomial of their associated links.
result Shows that the two polynomials are equivalent for brunnian θ-curves.
Solves equivalence problem for curves in G(2) flag varieties.
problem Equivalence problem for unparametrized curves in G(2)/P.
method Computes algebra of differential invariants for integral and generic curves.
result Provides a solution to the equivalence problem for curves in G(2) flag varieties.
Conformal Loxodromesmath.DG
Curved loxodromes on spheres are explained and their ODE derived.
problem Understanding curved analogues of compass-bearing curves on spheres.
method Explained curved loxodromes and derived the fifth order invariant ODE.
result Derived the fifth order invariant ODE for loxodromes.
New curves share invariant up to any fixed order.
problem Finite-order rondle invariants and lower central series.
method Formulation and construction of infinite sequences of curves.
result Infinitely many curves share invariants up to any fixed order.
Invariants count inflections and vertices in singular plane curves.
problem Counting inflections and vertices in singular plane curves.
method Defining invariants and to count inflections and vertices, respectively, and analyzing their properties.
result The invariants and are finite and bounded for curves without smooth components.
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…
Deep learning approximates geometric measures of planar curves.
problem Approximating differential invariants of planar curves.
method Utilizing deep neural networks to estimate geometric measures of planar curves.
result Deep neural networks can learn to overcome instabilities and sampling artifacts.
New invariants measure entanglement of open curves in 3D space.
problem Characterizing the complexity of open curves in 3-space.
method Introducing linkoids and virtual knots, and new invariants.
result Strong invariants of linkoids independent of virtual closure.
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
Invariant structures link to algebraic curves with specific properties.
problem Linking invariant hypercomplex structures to algebraic curves.
method Mapping invariant structures to algebraic curves with specific properties.
result Invariant hypercomplex structures correspond to algebraic curves with a flat projection and antiholomorphic involution.
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.
We give a complete description of all order 1 invariants of planar curves.
Extends Gromov invariant to Calabi-Yau 3-folds.
problem Counting embedded curves in Calabi-Yau 3-folds.
method Detailed study of bifurcations of moduli spaces of embedded pseudo-holomorphic curves.
result Integer-valued virtual count of embedded curves defined.
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…
We derive Frenet-type results and invariants of spatial curves immersed in -dimensional generalized Minkowski spaces, i.e., in linear spaces which satisfy all axioms of finite dimensional real Banach spaces except for the symmetry axiom. Further on, we characterize cylindrical helices and rectifying curves in such s…
This paper studies CR geometry of transversal curves in the 3-sphere.
problem Investigating CR geometry of transversal curves in the 3-sphere.
method Using local CR invariants of the 3-sphere, four global invariants are considered: phase anomaly, CR spin, Maslov index, and CR self-linking number.
result Closed critical curves of the simplest CR invariant variational problem for generic transversal curves are studied.
Formula derived for Gromov-Witten invariants of smooth curves.
problem Calculating Gromov-Witten invariants for smooth curves in genus zero.
method Closed formula derived from solving the degree zero limit of the loop equation for the complex projective line.
result Closed formula for generating series of Gromov-Witten invariants in genus zero.
Invariants count singularities and vertices of plane curves.
problem Counting singularities and vertices of plane curves.
method Defining and analyzing geometric invariants and .
result For almost all plane curves, these invariants are finite and bounded.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this artic…
The normalized Yamada polynomial is a polynomial invariant in variable A for theta-curves. In this work, we show that the coefficients of the power series obtained from this polynomial by the substitution A=e^x=1+x+x^2/2+x^3/6+... are finite-type invariants for theta-curves although the coefficients of original polynom…
Discussing moving frames for curve and surface invariants.
problem Identifying differential invariants of curves and surfaces.
method Using moving frames for Euclidean, affine, and conformal transformations.
result Determine differential invariants of curves and surfaces.
Given a Prym-Teichmüller curve in , this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the…
The paper studies properties of optimal metrics associated to curves on surfaces.
problem Investigating properties of optimal metrics associated to curves on surfaces.
method Starting from a filling curve and a separating curve, constructing a two integer parameter family of curves and deriving coarse length bounds and qualitative properties of their associated optimal metrics.
result There are infinitely many pairs of filling curves with distinct inf invariants but the same self-intersection number.
Study delta invariant of minimal generic curves on rational surfaces.
problem Recover delta invariant of curve germs from surface singularity topology.
method Explicit formulae for minimal generic curves on rational surfaces, proving delta invariant values for quotient singularities.
result Explicit formulae and values for delta invariant of minimal generic curves on rational surfaces.
In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group , a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) …
Continuing the program of math.SG/0012067 and math.SG/0310450, we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Tau…
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
Study how invariants change under bifurcations of curves.
problem Understanding how invariants of curves change under bifurcations.
method Analyzing , , , and invariants of curves under -bifurcations.
result Invariant changes under bifurcations, preserving its essential properties.
In this paper we construct effective invariants for braid monodromy of affine curves. We also prove that, for some curves, braid monodromy determines their topology. We apply this result to find a pair of curves with conjugate equations in a number field but which do not admit any orientation-preserving homeomorphism.
Hopf's Umlaufsatz relates the total curvature of a closed immersed plane curve to its rotation number. While the curvature of a curve changes under local deformations, its integral over a closed curve is invariant under regular homotopies. A natural question is whether one can find some non-trivial densities on a curve…
Study of small growth invariants in Goursat distributions.
problem Understanding local invariants of Goursat distributions.
method Analysis of the small growth sequence and its relation to structural invariants.
result Relate small growth invariants to structural invariants of Goursat distributions.
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…