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 \emph{\textbf{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 $I_k^c$ of combinatorial $k$-systoles of $S$ grows like $k$ and $\underset{k\to +\infty }{\mathrm{limsup}}(I_k^c-k)=+\infty$. 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.

12 pages, 8 figures