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 defined by a
-cocycle by extension was used
to show that the minimal number of sheets for the untwisted spun trefoil
is four.
The quandle
is an extension of
by a non-trivial
-cocycle with coefficient group
(hence as a set
).
More generally, it was proved that if a surface diagram
is colored non-trivially by
, 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 non-trivially, as the original tangle
has such a coloring.