An homotopy theorem for arrangements of double pseudolines
Luc Habert, Michel Pocchiola · 2006
We define a double pseudoline as a simple closed curve in the open Möbius band homotopic to the double of its core circle, and we define an arrangement of double pseudolines as a collection of double pseudolines such that every pair crosses in 4 points – the crossings being transversal – and induces a cell decomposition of the Möbius band whose 2-dimensional cells are 2-balls, except the unbounded cell which is a 2ball minus a point. Dual arrangements of boundaries of collection of pairwise disjoint 2-dimensional closed bounded planar convex sets are examples of arrangements of double pseudolines. We show that every pair of simple arrangements of double pseudolines is connected by a sequence of triangle-switches and that every simple arrangement of double pseudolines has a representation by a configuration of pairwise disjoint disks in the plane with pseudoline double tangents. This shows in particular that any double-permutation sequence of J.E. Goodman and R. Pollack (SoCG’05 page 159, [2]) has a representation by a configuration of pairwise disjoint disks in the plane with pseudoline double tangents. We also present some enumeration results for our arrangements, and a property of their subarrangements.