Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.
Max Neumann-Coto. "A characterization of shortest geodesics on surfaces." Algebr. Geom. Topol. 1 (1) 349 - 368, 2001. https://doi.org/10.2140/agt.2001.1.349