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

TL;DR

This paper investigates combinatorial $k$-systoles on punctured tori and pairs of pants, showing their maximal intersection numbers grow linearly with $k$ and addressing a combinatorial version of a conjecture related to geometric length.

Contribution

It establishes the growth rate of maximal intersection numbers of combinatorial $k$-systoles on specific surfaces, providing a positive answer to a combinatorial conjecture.

Findings

Maximal intersection number $I^c_k$ grows linearly with $k$

Limit superior of $I^c_k - k$ tends to infinity as $k$ increases

Results support the combinatorial version of the Erlandsson-Parlier conjecture

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 $I^c_k$ of combinatorial $k$-systoles of $S$ grows like $k$ and $\underset{k\rightarrow+\infty}{\limsup}(I^c_k-k)=+\infty$. 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.