Let $X$ be an elliptic K3 surface (over $\mathbb{C}$). Assume we have an elliptic fibration on $X$ that only has $I_1$ singular fibers.

If we fix a smooth fiber $F$ of such a fibration and a collection of paths to each of 24 singular fibers, then we will get 24 simple closed curves $C_1, \dots, C_{24}$ on $X$ --- the vanishing cycles (I do not know a proof for simplicity of these curves, but I think it can be done; anyway, in our situation we can make it primitive by dividing by an integer).

If we change the collection of paths by $i$-th Hurwitz move then, up to isotopy, the $i$-th vanishing cycle will become $\tau_{C_i}C_{i+1}$ and $i+1$-th vanishing cycle will become $C_i$ (where $\tau_C$ is a Dehn twist around $C$).

My question is: what configurations of isotopy classes (or, equivalently, homology classes) of simple closed curves on $F$ can be realized via this construction?


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