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

TL;DR

This paper investigates the conditions under which applying a Dehn twist to a curve on a closed surface results in a new curve at a specific distance in the curve graph, providing topological criteria and concrete examples.

Contribution

It establishes necessary and sufficient conditions for the distance between a curve and its Dehn twist image to be 4, and characterizes pairs at distances 5 or 6, with explicit examples.

Findings

Characterizes when the Dehn twist produces a distance 4 in the curve graph.

Provides topological conditions for distances 5 and 6 between curves.

Offers explicit example of curves at distance 5 with intersection number 144.

Abstract

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 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 $5 \leq d(a_0, T_{a_4}(a_0)) \leq 6$. 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 $144$.