Given two Hamiltonian isotopic curves in a surface, one would like to tell whether they are "close" or "far apart". A natural way to do that is to consider Hofer's metric which computes mechanical energy needed to deform one curve into the other. However due to lack of tools the large-scale Hofer geometry is only partially understood. On some surfaces (e.g. S^2) literally nothing is known.
The other possibility is to use a combinatorial metric where two curves are at distance one if they intersect at two points or less. This defines a metric on a family of isotopic curves by considering the shortest chain of elements at distance one. This construction can be seen as a symplectic analogue of curve complexes. In many aspects it behaves very similarly to Hofer's metric, at the same it is more accessible to direct computations with elementary tools.
I will discuss examples and techniques used to perform computations in both metrics.