We present some algorithms that provide useful topological information about curves in surfaces. One of the main algorithms computes the geometric intersection number of two properly embedded 1-manifolds $C_1$ and $C_2$ in a compact orientable surface $S$. The surface $S$ is presented via a triangulation or a handle structure, and the 1-manifolds are given in normal form via their normal coordinates. The running time is bounded above by a polynomial function of the number of triangles in the triangulation (or the number of handles in the handle structure), and the logarithm of the weight of $C_1$ and $C_2$. This algorithm represents an improvement over previous work, since its running time depends polynomially on the size of the triangulation of $S$ and it can deal with closed surfaces, unlike many earlier algorithms. Another algorithm, with similar bounds on its running time, can determine whether $C_1$ and $C_2$ are isotopic. We also present a closely related algorithm that can be used to place a standard 1-manifold into normal form.

翻译：我们提出了一系列算法，用于获取曲面中曲线的重要拓扑信息。其中主要算法之一可计算紧致可定向曲面$S$中两条恰当嵌入的一维流形$C_1$和$C_2$的几何交点数。曲面$S$通过三角剖分或柄体结构给出，而一维流形通过其法坐标呈现。该算法的运行时间上界是三角剖分中三角形数量（或柄体结构中柄体数）的多项式函数与$C_1$、$C_2$权重对数的乘积。该算法较以往工作有所改进，因为其运行时间依赖于$S$的三角剖分规模的多项式，并且与许多早期算法不同，它能处理闭曲面。另一个运行时间具有类似上界的算法可判断$C_1$与$C_2$是否同痕。我们还提出了一种密切相关的算法，可用于将标准一维流形化为正规形式。