  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.

Geometric Topology;Number Theory