Geometric intersections of loops on surfaces (Q2165568)

If we denote \(\simeq\) a free homotopy and \(\pitchfork\) a transverse intersection then, given two loops \(\varphi\) and \(\psi\) on a compact surface \(F\) we call \[I(\varphi,\psi)=\min\left\{\#(\phi'\cap \psi')\mid \phi'\simeq\phi, \psi'\simeq \psi, \phi'\pitchfork \psi'\right\}\] the geometric intersection number and \[SI(\varphi)=\min\left\{\# \mbox{ double points of } \varphi'\mid \varphi'\simeq \varphi, \varphi'\pitchfork\varphi \right\}\] the self-intersection number. In this work the authors obtain a systematic and straightforward method to compute geometric intersection numbers and self-intersection numbers of loops on oriented closed surfaces.