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 analogue 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 $X$ defined by a $2$-cocycle by extension was used to show that the minimal number of sheets for the untwisted spun trefoil is four. The quandle $X$ is an extension of $\mathbb{Z}_2[t]/(t^2+t+1)$ by a non-trivial $2$-cocycle with coefficient group $\mathbb{Z}_2$ (hence as a set $X=\mathbb{Z}_2 \times \mathbb{Z}_2[t]/(t^2+t+1)$). More generally, it was proved that if a surface diagram is colored non-trivially by $X$, then any broken surface diagram of the surface has at least four sheets.

An example, on the other hand, of a diagram of untwisted spun trefoil with exactly four sheets is easily constructed - the diagram obtained from the standard diagram of a trefoil tangle with two end points and three crossings, by spinning it around an axis that goes through the end points of the tangle, has four sheets, since the original tangle diagram has four arcs. It is also easy to color this diagram by $X$ non-trivially, as the original tangle has such a coloring.