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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.

          Ellipsoid
          (Image from Wikipedia.)

Q1. 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?

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.

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

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    $\begingroup$ How about a case when a pair you are looking for is infinitely close close pair of geodesics (Jacobi equation)? It could be a starting point in order to construct an example… $\endgroup$ Commented Mar 12, 2015 at 14:25

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The answer to Q1 is 'yes'. You can construct an explicit example as follows: Take a band $B$ around a geodesic $e$ (for 'equator') on the round sphere of radius 1, say, between two parallel circles above and below it. Now take the $n/2$-fold cover of this band, say, $\hat B$, and isometrically embed $\hat B$ into $\mathbb{R}^3$ as a surface of revolution and then smoothly close it up by attaching two disks, which can be done while keeping the result a genus $0$ surface $\Sigma$ of revolution. Now, the $n/2$-fold cover $\hat e$ of the geodesic $e$ will intersect all of the 'near-by' geodesics (which are also closed) exactly $n$ times. (Here, 'near-by' means geodesics that stay within $\hat B\subset\Sigma$.)

Q2 does not seem to me to be well-formulated.

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  • $\begingroup$ Great, Robert! Yes, Q2 is not well-formulated. I need to investigate papers such as Jean-François Le Gall's "Random geometry on the sphere" (arXiv abs link). $\endgroup$ Commented Mar 12, 2015 at 15:46
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    $\begingroup$ It seems that another example could be constructed by simply adding $n$ spikes along an equator of the sphere; the two curves that zigzag between the pikes and locally minimize length would have the required intersection. $\endgroup$ Commented Mar 12, 2015 at 16:15
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    $\begingroup$ @BenoîtKloeckner: That's very simple and clever! Is it clear that those zig-zags are geodesics? $\endgroup$ Commented Mar 12, 2015 at 18:27
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    $\begingroup$ @JosephO'Rourke: I think it's clear 'physically' that there will exist such geodesics if you place $n$ large bumps long the equator so that it's easier to go around them than over them. However, explicitly writing them down would be a challenge, I think. Benoît's example has the advantage that it's 'stable' in some sense because those geodesics should persist under small perturbations of the metric. On the other hand, my example produces a nonempty open set $U$ of closed geodesics on the sphere such that any two distinct geodesics in $U$ intersect in exactly $n$ points. $\endgroup$ Commented Mar 12, 2015 at 19:30
  • $\begingroup$ @BenoîtKloeckner & RobertBryant: Thanks to you both! Very insightful. $\endgroup$ Commented Mar 12, 2015 at 19:39

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