Distance 5 Curves in the Curve Graph of Closed Surfaces

Let [Formula: see text] denote a closed, orientable surface of genus [Formula: see text] and [Formula: see text] be the associated curve graph. Let [Formula: see text] be the path metric on [Formula: see text] and [Formula: see text] and [Formula: see text] be a pair of curves on [Formula: see text] with [Formula: see text]. In this article, we fix the vertex [Formula: see text] and apply the Dehn twist about [Formula: see text], [Formula: see text], to it in an attempt to create pairs of curves at a distance 5 apart. We give a necessary and sufficient topological condition for [Formula: see text] to be atleast 5. We also give an example of a pair of curves on [Formula: see text] which represent vertices at a distance 5 in [Formula: see text] with intersection number 144.