Drawing Permutations with Few Corners

TL;DR

This paper explores a geometric representation of permutations called tangles, composed of straight 45-degree segments, aiming to minimize crossings and corners simultaneously, with algorithms provided for specific permutation classes.

Contribution

It introduces a natural class of permutation representations called tangles and develops algorithms to minimize crossings and corners simultaneously for certain permutation classes.

Findings

Algorithms for constructing minimal-crossing, minimal-corner tangles.

Identification of permutation classes where both minimizations are achievable.

Enhanced understanding of geometric permutation representations.

Abstract

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.