Internship Proposal : Arrangements of Dp - Ribbons

Location. The internship will take place at the " Institut de Mathématiques de Jussieu-Paris Rive Gauche " in Jussieu. Scientific context and objectives. A DP-ribbon is a topological cylinder (a sphere with two boundaries) with a distinguished core circle with a distinguished side and an arrangement of DP-ribbons is a finite family of at least two DP-ribbons pairwise attached as shown in Fig. 1. 1 2 Figure 1. A DP-ribbon embedded in three-space (only the core circle is drawn, the distinguished side is indicated by small sky blue disks, and half-twists of the ribbon are indicated by horizontal dashed line segments), an arrangement of two DP-ribbons, and an indexed arrangement of two oriented DP-ribbons Note that the underlying surface of an arrangement of two DP-ribbons is a sphere with one crosscap and five boundaries. The genus of an arrangement of DP-ribbons is the genus of its underlying surface. The interest of this class of arrangements lies in the following three statements. Theorem 1 ([9]). The arrangements of DP-ribbons of genus 1 are exactly, modulo the adjunction of topological disks along their boundaries, the so-called arrangements of double pseudolines, i.e., the dual arrangements of finite families of pairwise disjoint convex bodies of (real two-dimensional) projective planes. Theorem 2 ([9]). An arrangement of DP-ribbons is of genus 1 if and only if its subarrangements of size 3, 4 and 5 are of genus 1.