Monday November 27th at 4.30pm

and spatial graph projection drawings.

A pseudoline arrangement is a finite set of curves in a plane, such that each one is homeomorphic to a line,

and two pseudolines always cross at one point. The advantage of pseudolines in comparison with (straight) lines

is that there exist combinatorial axiomatizations: an equivalence with rank 3 oriented matroids, and even a

first-order logical axiomatization.

Graph drawings whose edges are drawn with curves that cross at most once can be described with a similar but

extended logical structure. According to Ringel's theorem, two pseudoline arrangements in general positions can

always be transformed one into the other by a sequence of triangle flips. This result generalizes to complete

graph drawings.

Considering points in the 3-dimensional real space leads on one hand to a spatial graph formed by the straight

edges joining the points, and on the other hand to a rank 4 oriented matroid. With the previous results, will

see that this last combinatorial structure determines, here, for instance, a projection of the spatial graph up

to triangle flips.