# CONFLUENTES MATHEMATICI

Combinatorial $k$-systoles on a punctured torus and a pair of pants
Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 29-38.

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 ${I}_{k}^{c}$ of combinatorial $k$-systoles of $S$ grows like  $k$ and $\underset{k\to +\infty }{lim sup}\left({I}_{k}^{c}-k\right)=+\infty$.

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.

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.76
Classification: 32G15,  30F40
Keywords: closed geodesics, self-intersection, $k$-systole
ElHadji Abdou Aziz Diop 1; Masseye Gaye 1; Abdoul Karim Sane 1

1 Departement of Mathematics, Université Cheikh Anta Diop, Dakar, Senegal
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ElHadji Abdou Aziz Diop; Masseye Gaye; Abdoul Karim Sane. Combinatorial $k$-systoles on a punctured torus and a pair of pants. Confluentes Mathematici, Volume 13 (2021) no. 2, pp. 29-38. doi : 10.5802/cml.76. https://cml.centre-mersenne.org/articles/10.5802/cml.76/

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