Bouquets of curves in surfaces

Multiple curves on punctured orientable surfaces

We describe each multiple curve on the orientable surface of genus-$g$ with $n$ punctures and one boundary component by using this multiple curve's geometric intersection number with the embedded curves in this surface.

平面曲面精细曲线图的自同构

Automorphisms of fine curve graphs of planar surfaces

本文研究平面曲面精细曲线图的自同构问题，证明对至少7个穿刺点的无边界平面曲面，其精细曲线图的自同构群与曲面同胚群自然同构。

Uniform hyperbolicity for curve graphs of non-orientable 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 of (non-)orientable surfaces are 9-hyperbolic.

曲面上曲线的交叉数

Crossing number of curves on surfaces

该研究聚焦曲面上简单闭曲线系统的交叉数最小化问题，确定了亏格2曲面上最多12条曲线的最小交叉数，且证明最小化系统在曲面同胚与曲线同痕下唯一。

Finite Rigid Sets in Curve Complexes of Non-Orientable Surfaces

A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected non-orientable surfaces of genus $g$ with $n$ holes for $g+n \neq 4$.

Essential closed surfaces and finite coverings of negatively curved cusped 3-manifolds

The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.

极小亏格Seifert面上的素theta曲线

Prime theta-curves on minimal genus Seifert surfaces

研究极小亏格Seifert面上素结与本质弧的结合性质，证明其为素theta曲线。

Whitney's formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney's formula to curves on an oriented punctured surface. To define analogs of the rotation number and the index of a base point of a curve, we fix an arbitrary vector field on the surface. Similar formulas are obtained for non-based curves.

曲面及其轮廓曲线

Surfaces and their Profile Curves

本文研究$\mathbb{R}^3$中嵌入曲面纽结性与折叠曲线纽结性的关系，证明任意曲面可同痕变换为折叠曲线无环链环，并定义新不变量作为将曲面上固定曲线转化为折叠曲线的完全阻碍。

非可定向曲面中曲线的大 1-系统

Large 1-systems of Curves in Non-orientable Surfaces

本文将 Malestein-Rivin-Theran 的构造推广到非可定向曲面，给出了一般非可定向曲面中两两相交至多一次的曲线最大数量的下界。

Filling sets of curves on punctured surfaces

We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for odd k the orders of growth behave differently. We also study the corresponding questions when one requires that the curves be represented as systoles on hyperbolic complete finite area surfaces.

Ambient Surfaces and T-Fillings of T-Curves

T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latter by departing from non-convex polygons and we get a class of surfaces, the ambient surfaces, that we characterize topologically and on which T-curves are also well defined. Associated to each T-curve we construct a surface, the T-filling, which is topologically the analog of th…

Poisson algebras of curves on bordered surfaces and skein quantization

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quotient of the vector space spanned by the regular homotopy classes of curves on the bordered surface by generalizing the Goldman bracket and the Turaev cobracket. Moreover, we define a Poisson algebra of unoriented curves on a bordered surface and show that a quantization of the P…

Curves on surfaces, charts, and words

We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with appropriate weights. We also discuss relations between curves, Grothendieck dessins d'enfants, Riemann surfaces, Coxeter groups, and words.

Curves on surfaces and surgeries

We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries.

Geometric filling curves on surfaces

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

Vertices of closed curves in Riemannian surfaces

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.

从Legendrian对偶视角看双曲三维空间中带标架曲线的焦面与渐屈线

Focal surfaces and evolutes of framed curves in hyperbolic 3-space from the viewpoint of Legendrian duality

该研究针对双曲三维空间中可能带奇点的带标架曲线，借助Chen和Izumiya提出的Legendrian对偶性研究其焦面与渐屈线，给出相关对偶曲面的奇点分类，明确三类曲面的关联及奇点对偶关系。

Construction of Fractal Surfaces by Recurrent Fractal Interpolation Curves

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.