  Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\mathbf
b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$
determined by the pair $(\Delta,\,{\mathbf b})$. Under certain hypotheses, we
prove the equivalence between the fact that $(\Delta,\,{\mathbf b})$ is a mass
linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions.
I.} Int. Math. Res. Not. IMRN 8 (2010) 1506-1574.) and the vanishing of a
characteristic number of $E$. Denoting by ${\rm Ham}(M_{\Delta})$ the
Hamiltonian group of the symplectic manifold defined by $\Delta$, we determine
loops in ${\rm Ham}(M_{\Delta})$ that define infinite cyclic subgroups in
$\pi_1({\rm Ham}(M_{\Delta}))$, when $\Delta$ satisfies any of the following
conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii)
it is a $\Delta_p$ bundle over $\Delta_1$, (iii) $\Delta$ is the truncated
simplex associated with the one point blow up of ${\mathbb C}P^n$.
