An Experimental Study of Forbidden Patterns in Geometric Permutations by Combinatorial Lifting

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in ${\mathbb{R}}^3$. We show that this question, which is equivalent to deciding the emptiness of certain semi-algebraic sets bounded by cubic polynomials, can be “lifted” to a purely combinatorial problem. We propose an effective algorithm for that problem, and use it to gain new insights into the structure of geometric permutations.