Intersections of curves on surfaces
Intersections of curves on surfaces
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曲面上曲线的交点
DOI：
10.1007/bf02772960
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发表时间：
1985
影响因子：
1
通讯作者：
Peter Scott
中科院分区：
文献类型：
--
作者：
J. Hass;Peter Scott
AbstractThe作者认为曲面上的曲线有更多的交叉比最少可能在他们的同伦类。
定理1.设f是可定向曲面F上的一般位置弧或环，F与一个嵌入同伦但不嵌入。则F上存在一个嵌入的1-边形或2-边形，其边界为f的部分像。
定理2.设f是可定向曲面F上的一般位置弧或环，且F有多余自相交。然后，有一个奇异的1-gon或2-gon F上的F的图像的一部分所界定的。例子给出了类似的结果的情况下，两条曲线在一个表面上不举行，除了在众所周知的特殊情况下，当每个曲线是简单的。
AbstractThe authors consider curves on surfaces which have more intersections than the least possible in their homotopy class.
Theorem 1.Let f be a general position arc or loop on an orientable surface F which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on F bounded by part of the image of f.
Theorem 2.Let f be a general position arc or loop on an orientable surface F which has excess self-intersection. Then there is a singular 1-gon or 2-gon on F bounded by part of the image of f.Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.