The n-th crossing number of a graph G, denoted cr_n(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr_0(G) = a, cr_1(G) = b, and cr_2(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.
@article{arxiv.0911.0452,
title = {Unexpected behaviour of crossing sequences},
author = {Matt DeVos and Bojan Mohar and Robert Samal},
journal= {arXiv preprint arXiv:0911.0452},
year = {2010}
}