Distance $4$ curves on closed surfaces of arbitrary genus

Sreekrishna Palaparthi, Kuwari Mahanta

Published 2021-01-12Version 1

Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\mathcal{C}(S_g)$ the associated curve complex. The mapping class group of $S_g$ acts on $\mathcal{C}(S_g)$ by isometries. Since Dehn twists about certain curves generate the mapping class group of $S_g$, one can ask how Dehn twists move specific vertices in $\mathcal{C}(S_g)$ away from themselves. In this article, we answer this question for a specific case when the vertices are at a distance $3$. We show that if two curves represent vertices at a distance $3$ in $\mathcal{C}(S_g)$ then the Dehn twist of one curve about another yields two vertices at distance $4$. This produces many examples of curves on $S_g$ which are at distance $4$ in $\mathcal{C}(S_g)$. We also show that the minimum intersection number of any two curves at a distance $4$ on $S_g$, $i_{min}(g,4) \leq (2g-1)^2$.