On the crossing number of knots and links on surface in 3-manifolds

We give a lower bound of the minimum crossing number of knots and links projected on a 2-sided surface in a 3-manifold using the rank of fundamental groups which means that the crossing number can take arbitrary large values. On the contrary, we show that for a branched surface, the minimum crossing number is always zero. As an application, we give an upper bound of the three-page index by the crossing number.