Equivalent curves in surfaces
Christopher J. Leininger
本文从拓扑结构、代数结构和几何结构出发，研究了双曲曲面上曲线同伦类集合上的各种等价关系。这项工作的目的是确定这些等价之间的关系。
We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these equivalences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Self-intersection numbers of length-equivalent curves on surfaces
具有负欧拉示性数的可定向曲面中的两个自由同伦闭曲线类称为长度等价的，如果对曲面上的任意双曲结构，其中一类自由同伦闭曲线的测地线的长度等于另一类自由同伦闭曲线的测地线的长度.我们发现，有两个生成器的自由群中的元素是长度相等的，并具有不同的自相交数的元素在基本群的穿孔环面和元素在一条裤子。这一结果回答了关于长度等价类的公开问题，并提出了新的问题。
A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other class. Suppose $α$ and $β$ are two intersecting oriented closed curves on $F$ and $P$ and $Q$ are any two intersection points between them. If the two terms $\langleα*_Pβ\rangle$ and $\langleα*_Qβ\rangle$ in $[\langleα\rangle,\langleβ\rangle]$, the Goldman bracket between them, are t…
We examine an equivalence relation between free homotopy classes of closed curves on the pair of pants known as k-equivalence, a generalization of a concept previously defined by Leininger. We prove that two classes of closed curves on the pair of pants that are k-equivalent must also be 1-equivalent and 2-equivalent. We also examine properties of 1-equivalence on the pair of pants in greater depth.
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.
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。
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.
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。
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 $g+n \neq 4$.
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。
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.
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。
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。
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.
It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.
Ambient Surfaces and T-Fillings of T-Curves
T曲线是一种分段线性曲线，自20世纪90年代初以来，它已被成功地用于构造主要在实射影平面上具有预定拓扑的新的实代数曲线。事实上，T曲线可以用于任何真正的投影复曲面。我们在这里推广了后者的建设，离开非凸多边形，我们得到一类表面，周围的表面，我们的特征拓扑和T-曲线也很好地定义。与每一条T-曲线相关联，我们构造了一个曲面，即T-填充，它在拓扑上类似于实代数曲线通过复共轭的复化的商。然后利用T-填充证明了任意环境曲面上的T-曲线的一个定理，该定理类似于Harnack关于代数曲线的一个定理，并定义了T-曲线的可定向性和定向性，其定义方式与实代数中的定义方式相同。
We characterise when a set of simple closed curves in an orientable surface forms a bouquet, in terms of relations between the corresponding Dehn twists.
Surfaces proper homotopy equivalent to graphs and their Dehn-Nielsen-Baer maps
研究可能为无限型且边界非紧的第二可数可定向曲面与图恰当同伦等价的充要条件，探索相关映射类群间诱导映射的基本性质，以此为无限型曲面背景下德恩 - 尼尔森 - 贝尔类似物奠定基础。
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…
We bound the size of $d$-dimensional cubulations of finitely presented groups. We apply this bound to obtain acylindrical accessibility for actions on CAT(0) cube complexes and bounds on curves on surfaces.
Curves on surfaces, charts, and words
我们给出了定向曲面上的闭曲线的组合描述。我们用图表描述曲线的自同构，并计算用适当的权重计数的曲线的总数。我们还讨论了曲线之间的关系，格罗滕迪克dessins d'enfants，黎曼曲面，考克斯特群，和话。
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.
The bicorn curves on closed surfaces
本文研究了闭曲面上双耳曲线的双曲性质，并改进了有界测地像定理的界。
We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called $\mathbb{C}$-Fuchsian subgroups. In particular, if a Shimura surface contains one commensurability class of totally geodesic curves, it contains infinitely many.
Vertices of closed curves in Riemannian surfaces
我们揭示了完备黎曼曲面M的拓扑与最小顶点数之间的一些联系，即，M中闭曲线的测地曲率的临界点。特别地，我们证明了具有有限基本群的空间型是唯一的每个简单闭曲线都有两个以上顶点的曲面。进一步，我们把单连通空间型刻画为唯一的曲面，其中每一条包围紧浸入曲面的闭曲线都有两个以上的顶点。
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。
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.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。