Unexpected behaviour of crossing sequences

The n-th crossing number of a graph G, denoted c r 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 c r 0 ( G ) = a , c r 1 ( G ) = b , and c r 2 ( G ) = 0 . This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.