Distance $4$ curves on closed surfaces of arbitrary genus

Abstract: Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\mathcal{C}(S_g)$ be the associated curve complex. The mapping class group of $S_g$, $Mod(S_g)$ acts on $\mathcal{C}(S_g)$ by isometries. Since Dehn twists about certain curves generate $Mod(S_g)$, one can ask how Dehn twists move specific vertices in $\mathcal{C}(S_g)$ away from themselves. 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 tractable examples of distance $4$ vertices in $\mathcal{C}(S_g)$. We also show that the minimum intersection number of any two curves at a distance $4$ on $S_g$ is at most $(2g-1)^2$.