I'm not certain about terminology here so bear with me please.
If you put three nails in a flat surface, then wrap a rubber band around them, you will never get any parts of the band to cross. (The rubber band has to touch each nail exactly once. For simplicity, let's assume that no three nails will be colinear.)
However, with $n = 4$ nails, there seems to be a way to get one crossing, but not more than one, by tracing out the crossed quadrilateral. (Crossings right by the nails don't count — think of nails as points on a plane.)
With five nails it seems like I can get up to five crossing if I trace out the pentagram. But can I get more crossings? For example, the (7/2) heptagram gives me 7 crossings, but I can also get 14 crossings using only 7 nails if I trace out the (7/3) heptagram.
In general, what is the name of these crossings, and can their number be easily calculated from the number of vertices (nails)?
Grünbaum gave an explicit quasipolynomial formula for the maximal number $a(n)$ of self-intersections of an $n$-gon, $n \geq 3$:
$$a(n) = \left\{ \begin{array}{ll} \frac{n(n-3)}{2}, &\textrm{$n$ odd}\\ \frac{n(n-4)}{2}+1, &\textrm{$n$ even}. \end{array}\right. ;$$ the first terms are: $$0, 1, 5, 7, 14, 17, 27, 31, 44, 49, \ldots .$$ This sequence is OEIS A105638, Maximum number of intersections in self-intersecting $n$-gon.