  A permutation may be represented by a collection of paths in the plane. We
consider a natural class of such representations, which we call tangles, in
which the paths consist of straight segments at 45 degree angles, and the
permutation is decomposed into nearest-neighbour transpositions. We address the
problem of minimizing the number of crossings together with the number of
corners of the paths, focusing on classes of permutations in which both can be
minimized simultaneously. We give algorithms for computing such tangles for
several classes of permutations.
