a boundary-fixing diffeomorphisms from the cylinder to itself composed with a mapping from the cylinder to $S^{2}$ taking the boundaries of the cylinder to $\dot{\gamma}$ and to $-\dot{\gamma}$ and mapping the line $s\times\{i\}$, $i\in \mathbf{I}$ to a great circle from $\dot{\gamma}(s)$ to $-\dot{\gamma}(s)$ for each $s\in S^{1}$. Let us fix the second map in this composition and call it $g$. By [@Gra] there is a $\mathbb{Z}$-worth of boundary-preserving diffeomorphisms of the cylinder $S^{1}\times \mathbf{I}$ to itself up to homotopy by such diffeomorphisms. The generator of the this homotopy group is a Dehn twist about a boundary-parallel curve.

2.  Let $p_{s}$ be the image of $s\times {I}$ in the $S^{2}$ under a power of the generator which we found in step 1. Let $D_{\varepsilon}$ be a disc in $S^{2}$ of radius $\varepsilon$ centred at the image of $s\times \{1\}$. The homotopy which fixes $s\times \{0\}$ and $s\times\{1\}$ and revolves $p_{s}\cap\partial D_{\varepsilon}$ in a full circle around $\partial D_{\varepsilon}$ for all $s\in S^{1}$ undoes the Dehn twist.

    Concretely, let’s look at $\varphi_{t}$ $t\in \mathbf{I}$ which takes $p_{s}$ to a smoothing of the curve obtained by taking a radius of $D_{\varepsilon}$ at angle $2\pi t$, then travelling around the circumference of $D_{\varepsilon}$ clockwise 