Combinatorial k-systoles on a punctured torus and a pair of pants

Abstract

In this paper S denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of combinatorial k-systoles, that is closed curves with self-intersection numbers greater than k and with least combinatorial length. We show that the maximal intersection number Ic k of combinatorial k-systoles of S grows like k and lim sup k→+∞ (Ic k − k) = +∞. This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson-Parlier conjecture, originally formulated for the geometric length.