An homotopy theorem for arrangements of double pseudolines

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. γ γ γ γ′ γ γ γ γ γ M(γ) l Figure 1: The Möbius band and its core circle, a (monotone) double pseudoline γ and the Möbius band M(γ) bounded by γ, an arrangement of two double pseudolines γ and γ′, a collection of two double pseudolines with 4 crossing points but which is not an arrangement because the cell intersection of the associated Möbius bands is not a 2-ball, and a triangle-switch. 1. Arrangements of double pseudolines in the Möbius band. Let M be the open Möbius band, say quotient of R under the map ι : R → R that assigns to the pair (θ, u) the pair (θ + π,−u), and let c : [0, 1] → M, c(t) = (πt, 0), be its core circle. A pseudoline in M is a simple closed path in M homotopic to its core circle and a double pseudoline in M Department of Computer Science, Ecole normale supérieure, Paris, {Luc.Habert,Michel.Pocchiola}@ens.fr is a simple closed path in M homotopic to the double cc of its core circle. Boundary curves of tubular neighborhoods of pseudolines are examples of double pseudolines. We define an arrangement of double pseudolines in M as a finite collection Γ of double pseudolines in M such that every pair of elements of Γ crosses in 4 points – the crossings being transversal – and induces a cell decomposition of M whose 2-dimensional cells are 2-balls, except the unbounded cell which is a 2-ball minus a point, and we define the chirotope χΓ of Γ as the map that assigns to each γ ∈ Γ and to each ordered triple u, v, w of vertices lying on γ of the cell decomposition XΓ of M induced by the elements of Γ the value +1 if walking along the curve γ we encounter the vertices in cyclic order uvwuvw · · · ; −1 otherwise. An arrangement of double pseudolines is called simple if exactly two double pseudolines meet at every vertex of the induced cell decomposition of M. A simple arrangement of double pseudolines is called thin if there is no vertex of the induced cell decomposition lying in the interiors of the closed Möbius bands bounded by the double pseudolines. Thin arrangements are obtained from simple (finite) arrangements of pseudolines by replacing the pseudolines by the boundary curves of suitable tubular neighborhoods. Collections of dual curves of boundaries of pairwise disjoint 2-dimensional closed bounded planar convex sets are examples of arrangements of double pseudolines; these arrangements are simple and thin under the additional assumption that there is no line transversal to three convex sets; these arrangements are also monotone with respect to the core circle in the sense every meridian θ = c of the Möbius band crosses every double pseudoline exactly twice. Finally we observe that, as in the case of arrangements of pseudolines, the set of arrangements of double pseudolines is stable by triangle-switch. The main result of the paper is the following. Theorem 1 Let Γ be a simple arrangement in M of double pseudolines, X the induced cell decomposition of M, and γ ∈ Γ. Assume that there is a vertex of X lying in the interior of the Möbius band M(γ) bounded by γ. Then there is a triangular face of X The dual of a planar smooth curve is the curve in the space of undirected lines of the plane of the tangent lines to the curve. We identify the space of undirected lines of the plane with the Möbius band M via the map that assigns to the pair (θ, u) the line with equation u− x sin θ + y cos θ = 0.