We prove that, as $m$ grows, any family of $m$ homotopically distinct closed curves on a surface induces a number of crossings that grows at least like $(m \log m)^2$. We use this to answer two questions of Pach, Tardos and Toth related to crossing numbers of drawings of multigraphs where edges are required to be non-homotopic. Furthermore, we generalize these results, obtaining effective bounds with optimal growth rates on every orientable surface.

翻译：我们证明，随着$m$的增长，曲面上任意$m$条同伦相异的闭曲线族所诱导的交叉数至少以$(m \log m)^2$的速度增长。利用这一结论，我们回答了Pach、Tardos和Toth提出的两个关于多重图绘制交叉数的问题，其中要求图的边必须是非同伦的。此外，我们推广了这些结果，在每一个可定向曲面上得到了具有最优增长率的有效界。