Is there a possibility to analytically determine how often a Lissajous curve on the surface of a sphere crosses itself?
The Lissajous figure (on the unit sphere) is given by:
x = $ \sin(n_{\theta} t) \cos(n_{\phi} t) $
y = $ \sin(n_{\theta} t) \sin(n_{\phi} t) $
z = $ \cos(n_{\theta} t) $
where $n_{\theta}$ and $n_{\phi}$ are arbitrary integers and $t \in [0, 2\pi)$.
An example for such a trajectory on a sphere with $n_{\phi}=21$ and $n_{\theta}=20$ is given in this image: 
I think I found the answer this arxiv.org article. The number of crossing points is:
$ n_{\theta} ~( n_{\phi} -1) +2 $
The two poles of the sphere are traversersed $ n_{\theta}$ times while $t$ runs from $0$ to $\pi$. The remaining points are travesed two times each.
The poles are obviously two crossing points. If you want to consider each pair of arcs separately, there are obviously $n_\theta(n_\theta+1)$ polar crossings.
Having made some sketches, I think there are simply $n_\theta n_\phi$ non-polar crossings, for a total of $n_\theta (n_\phi + n_\theta + 1)$.