Example of distance $5$ curves on closed surfaces

Let $S_g$ denote a closed, orientable surface of genus $g \geq 2$ and $\mathcal{C}(S_g)$ be the associated curve graph. Let $d$ be the path metric on $\mathcal{C}(S_g)$ and $a_0$ and $a_4$ be two curves on $S_g$ with $d(a_0, a_4) = 4$. It follows from the triangle inequality that $d(a_0, T_{a_4}(a_0)) \leq 6$. In this article we give a criterion for when $d(a_0, T_{a_4}(a_0)) = 4$ and when $d(a_0, T_{a_4}(a_0)) \geq 5$. We further give an explicit example of a pair of curves on $S_2$ which represent vertices at a distance $5$ in $\mathcal{C}(S_2)$.Comment: 21 pages, 39 figures, 1 tabl