Sweeping $x$-monotone pseudolines

We study the problem of sweeping a pseudoline arrangement with $n$ $x$-monotone curves with a rope (an $x$-monotone curve that connects the points at infinity). The rope can move by flipping over a face of the arrangement, replacing parts of it from the lower to the upper chain of the face. Counting as length of the rope the number of edges, what rope-length can be needed in such a sweep? We show that all such arrangements can be swept with rope-length at most $2n-2$, and for some arrangements rope-length at least $7(n-2)\mathrm{/}4+1$ is required. We also discuss some complexity issues around the problem of computing a sweep with the shortest rope-length.