Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$. $S$ has at least three simple (non-self-intersecting), closed geodesics by a theorem of Lyusternik and Shnirel'man. Alternatively, any Riemannian metric on $\mathbb{S}^2$ leads to at least three simple, closed geodesics.

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. Is it true that, for every even $n > 2$, there is an $S$ that has a pair of simple, closed geodesics that cross each other $n$ times?Q1

On an ellipsoid, each pair of the simple, closed geodesics cross twice.
This feels like the generic situation.
But my guess is that the answer to Q1 is *Yes*; but an explicit construction
is eluding me.

. Is there any sense in which $n=2$ crossings is the usual situation?Q2