Twist numbers on hyperbolic once-punctured tori

On a hyperbolic surface homeomorphic to a torus with a puncture, each oriented simple geodesic inherits a well-defined relative twist number in $[0,1]$, given by the ratio to its hyperbolic length of the hyperbolic distance between the orthogonal projections of the cusp (or boundary) on its left and right, respectively. With the Markov Uniqueness Conjecture in mind, the twist numbers for simple geodesics on the modular torus $\mathcal{X}$ are of particular interest. Up to isometries of $\mathcal{X}$, simple geodesics are parameterized naturally by $\mathbb{Q}\cap[0,1]$, and the relative twist number yields a map $\tau_{\mathcal{X}}:\mathbb{Q}\cap [0,1] \to [0,1]$. We use hyperbolic geometry and the Farey graph to show that the graph of $\tau_{\mathcal{X}}$ is dense in $[0,1]\times [0,1]$, and the same conclusion holds for any complete hyperbolic structure $\mathcal{Y}$ on the punctured torus. It follows that the twist number of a simple closed curve on the punctured torus does not extend continuously to the space of measured laminations. We also include some explicit calculations of geometric quantities associated to Markov triples, and the curious fact that $\tau_{\mathcal{X}}$ is never equal to zero.