论文标题
三角网格台球和知识图
Triangular-Grid Billiards and Plabic Graphs
论文作者
论文摘要
考虑到三角形网格中的多边形$ p $，我们通过天然台球系统获得置换$π_p$，在$ p $的内部，光线弹跳。 $π_p$中的不同循环对应于光束的不同轨迹。我们证明\ [\ text {afore}（p）\ geq 6 \ text {cyc}（p）-6 \ quad \ text {and} \ quad \ quad \ text {perim}（p）\ geq \ geq \ geq \ frac {7} {7} $ \ text {afir}（p）$和$ \ text {perim}（p）$分别是（适当归一化的）面积和$ p $的周长，$ \ text {cyc}（p）$是$π_p$中的循环数。有关$ \ text {afore}（p）$的不等式很紧，我们表征了满足$ \ text {aind}（p）= 6 \ text {cyc}（p）-6 $的多边形$ p $。这些结果可以用如下的尼科夫的波现图进行重新重新制定。让$ g $为连接的缩减的尺寸$ 2 $。假设$ g $具有$ n $标记的边界点和$ v $（内部）顶点，而让$ c $为旅行置换$ g $的周期数量。然后，我们有\ [v \ geq 6c-6 \ quad \ text {and} \ quad n \ geq \ frac {7} {2} {2} c- \ frac {3} {2} {2} {2}。\]
Given a polygon $P$ in the triangular grid, we obtain a permutation $π_P$ via a natural billiards system in which beams of light bounce around inside of $P$. The different cycles in $π_P$ correspond to the different trajectories of light beams. We prove that \[\text{area}(P)\geq 6\text{cyc}(P)-6\quad\text{and}\quad\text{perim}(P)\geq\frac{7}{2}\text{cyc}(P)-\frac{3}{2},\] where $\text{area}(P)$ and $\text{perim}(P)$ are the (appropriately normalized) area and perimeter of $P$, respectively, and $\text{cyc}(P)$ is the number of cycles in $π_P$. The inequality concerning $\text{area}(P)$ is tight, and we characterize the polygons $P$ satisfying $\text{area}(P)=6\text{cyc}(P)-6$. These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let $G$ be a connected reduced plabic graph with essential dimension $2$. Suppose $G$ has $n$ marked boundary points and $v$ (internal) vertices, and let $c$ be the number of cycles in the trip permutation of $G$. Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]
