a measure of the amount of twisting around the boundary effected by $\phi$ compared to a “canonical”—e.g., pseudo-Anosov—representative of its (free) isotopy class. More precisely, consider the image of $\phi$ under the natural map ${\mathrm{Aut}}(S,\partial S)\rightarrow {\mathrm{Aut}}(S)$ which drops the requirement that an isotopy fixes the boundary pointwise. In this latter group, $\phi$ is isotopic to its Nielsen-Thurston representative; that is, there is an isotopy $\Phi:S\times [0,1]\rightarrow S$ such that $\Phi_0=\phi$ and $\Phi_1$ is either periodic, reducible, or pseudo-Anosov[^1]. Considering the restriction of $\Phi$ to the boundary, we obtain a homeomorphism: $$\Phi_\partial: \partial S \times[0,1]  \rightarrow \partial S\times [0,1]$$ defined by $\Phi_\partial(x,t)=(\Phi_t(x),t)$. The fractional Dehn twist coefficient $\tau(\phi)$ can be defined as the winding number of the arc $\Phi(\theta\times[0,1])$ where $\theta\in \partial S$ is a basepoint[^2]. This would appear only to associate a real number to $\phi$, which could depend on the choice of basepoint and isotopy. The Nielsen-Thurston classification, however, shows that this winding number is a well-defined rational-valued invariant $\tau(\phi)\in {\mathbb{Q}}$. The definition extends easily to surfaces with several boundary circles, in which case there is a corresponding twist coefficient for each component of the boundary. Here we will be concerned only with the case of connected boundary.

