Distance $5$ Curves in the Curve Graph of Closed Surfaces

Let $S_g$ denote a closed, orientable surface of genus $g\ge 2$ 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 a pair of curves on $S_g$ with $d(a_0,a_4)=4$. In this article, we fix the vertex $a_0$ and apply the Dehn twist about $a_4$, $T_{a_4}$, to it in an attempt to create pairs of curves at a distance $5$ apart. We give a necessary and sufficient topological condition for $d(a_0,T_{a_4}(a_0))$ to be $4$. We then characterise the pairs of $a_0$ and $a_4$ for which $5\le d(a_0,T_{a_4}(a_0))\le 6$. Lastly, we give an example of a pair of curves on $S_2$ which represent vertices at a distance $5$ in $\mathcal{C}(S_2)$ with intersection number $144$.