An experimental study of forbidden patterns in geometric permutations by combinatorial lifting

Abstract

We study the problem of deciding if a given triple of permutations can be realized as geometric permutations of disjoint convex sets in ℝ3. 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.