Untangling two systems of noncrossing curves

Abstract

We consider two systems of curves $(\alpha_1,...,\alpha_m)$ and $(\beta_1,...,\beta_n)$ drawn on a compact two-dimensional surface $M$ with boundary. Each $\alpha_i$ and each $\beta_j$ is either an arc meeting the boundary of $M$ at its two endpoints, or a closed curve. The $\alpha_i$ are pairwise disjoint except for possibly sharing endpoints, and similarly for the $\beta_j$. We want to "untangle" the $\beta_j$ from the $\alpha_i$ by a self-homeomorphism of $M$; more precisely, we seek a homeomorphism $\phi:M\rightarrow M$ fixing the boundary of $M$ pointwise such that the total number of crossings of the $\alpha_i$ with the $\phi(\beta_j)$ is as small as possible. This problem is motivated by an application in the algorithmic theory of embeddings and 3-manifolds. We prove that if $M$ is planar, i.e., a sphere with $h\geq 0$ boundary components ("holes"), then $O(mn)$ crossings can be achieved (independently of $h$), which is asymptotically tight, as an easy lower bound shows. In general, for an arbitrary (orientable or nonorientable) surface $M$ with $h$ holes and of (orientable or nonorientable) genus $g$, we obtain an $O((m+n)^4)$ upper bound, again independent of $h$ and $g$. The proofs rely, among others, on a result concerning simultaneous planar drawings of graphs by Erten and Kobourov.