arXiv research

Study of orbifold mapping class groups via arc and curve actions.

The action of the mapping class group of a surface on the collection of homotopy classes of disjointly embedded curves or arcs in the surface is discussed here as a tool for understanding Riemann's moduli space and its topological and geometric invariants. Furthermore, appropriate completions, elaborations, or quotient…

The paper shows how to rearrange arcs to form closed curves.

Homotopy types of curve and arc complexes are studied.

The grand arc graph's asymptotic dimension is shown to be infinite.

The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

Study proves hyperfiniteness of mapping class group actions on surface graphs.

In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian $\mathbb{R}^4$, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere…

Study arcs on surfaces, focusing on topological aspects and group actions.

We provide a new angle and obtain new results on a class of metrics on length-normalized curves in $d$ dimensions, represented by their unit tangents expressed as a function of arc-length, which are functions from the unit interval to the $(d-1)$-dimensional unit sphere. These metrics are derived from the combined acti…

Three methods solve spatial rational curves with rational arc length.

We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Euc…

In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner product is preserved by the full symmetry group of Euclidean space (translations, rotations, and reflections). In equiaffine ge…

Proves prime theta-curves for knots on minimal genus surfaces.

Sharp chord-arc estimates for curve shortening flow on spheres.

New method identifies vanishing arcs for curve singularities.

Curve diffusion flow straightens curves with endpoints on intersecting lines.

Study on curve shortening flow with boundary conditions, proving convergence or contraction.

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natu…

We show that the arc graph of $S_g^1$ is a coarse Lipschitz retract of the free splitting complex of $F_{2g}$. We also show that the arc and curve graph of $S_g^1$ is a coarse Lipschitz retract of both the cyclic splitting graph of $F_{2g}$ and the maximally cyclic splitting graph of $F_{2g}$.

We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured sur…

Self-affine arcs without inner weak separation are parabolic segments.

Shifts are not type-preserving on surface graphs.

Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.

Researchers describe the Gromov boundary of a graph related to surfaces.

Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs …

A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…

The paper generalizes rectifying and normal curves in Lorentzian n-space.

Paper estimates optimal ROC curve arc length and AUC, improving classification performance.

In "Tunnel one, fibered links", the second author showed that the tunnel of a tunnel number one, fibered link can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we analyze how the arc behaves under the monodromy action, and show that the tunnel arc is nearly clean, with t…

The method for approximation of planar curve by circular arcs with length preservation, proposed by I.Kh. Sabitov and A.V. Slovesnov, is analyzed. We extend the applicability of the method, and consider some corollaries, not related to the approximation problem. Inequalities for the length of a convex spiral arc with p…

New graphs show hierarchical hyperbolic properties, extending previous work.

We prove that on a punctured oriented surface with Euler characteristic chi < 0, the maximal cardinality of a set of essential simple arcs that are pairwise non-homotopic and intersecting at most once is 2|chi|(|chi|+1). This gives a cubic estimate in |chi| for a set of curves pairwise intersecting at most once on a cl…

Defines a new family of curves in space with applications.

The paper introduces various canonical parameterizations for 2D-curved shapes.

This paper constructs PH spline curves with prescribed arc lengths.

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil kno…

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a confor…

We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in Mio-Srivastava-Joshi on the space of arc-length parameterized loops. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of …

We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this init…

Infinite type surfaces can be perfectly divided into triangles.

We extend the notion of unicorn paths between two arcs introduced by Hensel, Przytycki and Webb to the case where we replace one arc with a geodesic asymptotic to a lamination. Using these paths, we give new proofs of the results of Klarreich and Schleimer identifying the Gromov boundaries of the curve graph and the ar…

Connectedness proved for $\mathbb{Z}^d$ actions on 1D manifolds by $C^2$ diffeomorphisms.

Given any closed, connected, orientable $3$--manifold and integers $g\geq g(M), D > 0$, we show the existence of knots in $M$ whose genus $g$ bridge number is greater than $D$. These knots lie in a page of an open book decomposition of $M$, and the proof proceeds by examining the action of the map induced by the monodr…

Quadratic growth of intersecting curves on surfaces resolved.

We propose to describe the variety of galaxies from SDSS by using only one affine parameter. To this aim, we build the Principal Curve (P-curve) passing through the spine of the data point cloud, considering the eigenspace derived from Principal Component Analysis of morphological, physical and photometric galaxy prope…

Study of tree automorphisms via arc-stabilizers.