多边形的莫比乌斯变换和三维空间的划分
Mobius transformations of polygons and partitions of 3-space
摘要：
多边形纽结K在R^3（并无穷大）的球面反演下的像是由圆弧组成的简单闭曲线，与K的镜像具有相同的纽结类型。假设我们用直线重新连接倒多边形的顶点，形成一个新的多边形。这可能是不同的结类型。例如，通过选择不同的反转球体，可以将某个7段8字形结转换为8字形结、右手和左手三叶形结或解结。在这个过程中，从给定的原始多边形K可以得到哪些节点类型？我们发现，对于大的n，几乎所有的n段结类型不能从一个初始的n段多边形，使用一个单一的反演，甚至整个莫比乌斯群。由一个n顶点多边形产生的纽结类型的数目由R^3中某个n（n-3）/2个圆2-球面系统的补域的数目所限定。我们证明了域的个数至多是球面个数的多项式。在分析中，我们得到了一个精确的公式的互补域的数目。另一方面，可以由n段多边形表示的结类型的数量是n的指数。
英文摘要：
The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K. Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon. This may be a different knot type. For example, a certain 7-segment figure-8 knot can be transformed to a figure-8 knot, right and left handed trefoils, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, almost all n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Mobius group. The number of knot types arising from an n-vertex polygon is bounded by the number of complementary domains of a certain system of n(n-3)/2 round 2-spheres in R^3. We show the number of domains is at most polynomial in the number of spheres. In the analysis, we obtain an exact formula for the number of complementary domains. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.
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