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：我们提出了一些算法，这些算法能提供关于曲面中曲线的有用拓扑信息。其中一个主要算法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$C_2$的几何相交数。曲面$S$通过三角剖分或手柄结构给出，而一维流形则通过其正规坐标以正规形式表示。该算法的运行时间上界是关于三角剖分中三角形数量（或手柄结构中的手柄数）以及$C_1$和$C_2$权重对数的多项式函数。与许多早期算法不同，该算法运行时间多项式依赖于$S$的三角剖分规模，且能处理闭曲面，这体现了对先前工作的改进。另一个具有相似运行时间上界的算法可判定$C_1$和$C_2$是否同痕。我们还提出一个密切相关的算法，可用于将标准一维流形化为正规形式。