Abstract: Three numbers can be associated to a deformation class of closed curves on a surface S:
- the self-intersection number (this is the smallest number of times a representative of the curve crosses itself),
- the word length (that is the smallest number of letters, in a certain alphabet chosen apriori, one needs to describe the class ) and
- the length of the geodesic in the class (this is the length of the shortest representative of the class, where the way of computing length is chosen beforhand).
The interrelations of these three numbers exhibit many patterns when explicitly determined or approximated using algorithms and a computer. We will discuss how these computations can lead to counterexamples of existing conjectures, to the discovery of new conjectures and to subsequent theorems in some cases.