论文标题
具有固定顶点角度的凸面平面多边形的双循环多型
Dual-cyclic polytopes of convex planar polygons with fixed vertex angles
论文作者
论文摘要
如果我们将角度固定在凸面平面$ n $ gon的顶点，则其边缘的长度必须满足两个线性约束，才能关闭它。如果我们还需要单位周边，我们的$ n $ edge长度的向量形成了尺寸$ n-3 $的凸多角形，每个方面都由那些$ n $ gons组成，其中特定边缘的长度下降到零。 Bavard和Ghys需要单位区域，这使它们具有双曲线。这两个多面体在组合上是等效的，因此对于我们的目的而言，要么可以。 当角度良好时，这种固定角度的多层层在组合上会更丰富。我们说，固定的外角是“多数派的”，当连续一半以上的连续字符串总计为$π$以上。当$ n $是奇数时，我们表明，任何多数主导角度的固定角度多层与环状polytope $ c_ {n-3}（n）$都是双重的。为了将结果扩展到$ n $，我们要求该角度还具有“偶极子束缚”：$ n $ n $ n $ n $ n/2 $ sums的$ n $ strings copisisy $π$，而$ n/2 $的总和超过$π$重叠的$ n/2 $尽可能尽可能多地包含特定的角度。 但是，固定顶点的角度并不常见。人们经常固定边缘长度。这很难，部分原因是固定长度$ n $ - gons可能不是凸，但主要是因为固定长度非线性地约束角度 - 因此，所得的模量空间（称为“多边形空间”）是弯曲的。 Kapovich和Millson使用Schwarz-Christoffel Maps，表明该$ n $ gons是凸面的多边形空间的子集，并且逆时针逆时针与上面的固定角式逆时针呈同型，对于相同的固定值。因此，每个这样的子集都是拓扑多层。而且，每当固定长度占主导地位时，它都是双循环的，甚至对于$ n $，都具有偶极子票。
If we fix the angles at the vertices of a convex planar $n$-gon, the lengths of its edges must satisfy two linear constraints in order for it to close up. If we also require unit perimeter, our vectors of $n$ edge lengths form a convex polytope of dimension $n-3$, each facet of which consists of those $n$-gons in which the length of a particular edge has fallen to zero. Bavard and Ghys require unit area instead, which gives them a hyperbolic polytope. Those two polytopes are combinatorially equivalent, so either is fine for our purposes. Such a fixed-angles polytope is combinatorially richer when the angles are well balanced. We say that fixed external angles are "majority dominant" when every consecutive string of more than half of them sums to more than $π$. When $n$ is odd, we show that the fixed-angles polytope for any majority-dominant angles is dual to the cyclic polytope $C_{n-3}(n)$. To extend that result to even $n$, we require that the angles also have "dipole tie-breaking": None of the $n$ strings of length $n/2$ sums to precisely $π$, and the $n/2$ that sum to more than $π$ overlap as much as possible, all containing a particular angle. Fixing the vertex angles is uncommon, however; people more often fix the edge lengths. That is harder, in part because fixed-lengths $n$-gons may not be convex, but mostly because fixing the lengths constrains the angles nonlinearly -- so the resulting moduli spaces, called "polygon spaces", are curved. Using Schwarz-Christoffel maps, Kapovich and Millson show that the subset of that polygon space in which the $n$-gons are convex and traversed counterclockwise is homeomorphic to the fixed-angles polytope above, for those same fixed values. Each such subset is thus a topological polytope; and it is dual cyclic whenever the fixed lengths are majority dominant and, for even $n$, have dipole tie-breaking.
