Self-intersection numbers of length-equivalent curves on surfaces

Two free homotopy classes of closed curves in an orientable surface with negative Euler characteristic are said to be length-equivalent if the lengths of the geodesics in the two classes are equal for every hyperbolic structure on the surface. We show that there are elements in the free group on two generators that are length-equivalent but have different self-intersection numbers. This applies to both the punctured torus and the pair of pants. Our result answers open questions about length-equivalence classes and raises new ones.