论文标题
距离$ 5 $曲线的曲线曲线图
Distance $5$ Curves in the Curve Graph of Closed Surfaces
论文作者
论文摘要
令$ s_g $表示属于$ g \ geq 2 $和$ \ mathcal {c}（s_g）$的封闭式，可定向的表面为关联的曲线图。令$ d $为$ \ m varycal {c}（s_g）$和$ a_0 $和$ a_4 $的路径度量是$ s_g $的一对曲线，$ s_g $ with $ d（a_0，a_4）= 4 $。在本文中，我们将顶点$ a_0 $修复，并将dehn twist应用于$ a_4 $，$ t_ {a_4} $，以尝试以$ 5 $相距的距离创建一对曲线。我们给出了$ d（a_0，t_ {a_4}（a_0））$的必要且充分的拓扑条件，为$ 4 $。然后，我们表征$ a_0 $和$ a_4 $的对，$ 5 \ leq d（a_0，t_ {a_4}（a_0））\ leq 6 $。最后，我们给出了$ s_2 $上的一对曲线的示例，该曲线代表距离$ 5 $ in $ \ mathcal {c}（s_2）$的顶点$ 144 $。
Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\mathcal{C}(S_g)$ be the associated curve graph. Let $d$ be the path metric on $\mathcal{C}(S_g)$ and $a_0$ and $a_4$ be a pair of curves on $S_g$ with $d(a_0, a_4) = 4$. In this article, we fix the vertex $a_0$ and apply the Dehn twist about $a_4$, $T_{a_4}$, to it in an attempt to create pairs of curves at a distance $5$ apart. We give a necessary and sufficient topological condition for $d(a_0, T_{a_4}(a_0))$ to be $4$. We then characterise the pairs of $a_0$ and $a_4$ for which $5 \leq d(a_0, T_{a_4}(a_0)) \leq 6$. Lastly, we give an example of a pair of curves on $S_2$ which represent vertices at a distance $5$ in $\mathcal{C}(S_2)$ with intersection number $144$.
