Minimal number of broken sheets in knotted surface diagrams

For a classical knot, the number of crossings and the number of arcs coincide. For knotted surfaces, then, another analog of the crossing number is the minimal number of broken sheets that are needed to form a diagram of a given knotted surface.

In [SaiSat03*], a quandle defined by a $2$-cocycle was used to show that the minimal number of sheets for the untwisted spun trefoil is four.