= Solution
Choose an essential simple closed curve $c$ in the oriented <topological surface> and an annular neighborhood $N(c)\cong S^1\times[0,1]$. A positive <Dehn twist> about $c$ is represented on this annulus by
$$
\boxed{\tau_c(e^{2\pi i\theta},s)=(e^{2\pi i(\theta+\rho(s))},s),}
$$
where $\rho$ is zero near $0$, one near $1$, and increases between them. Since a full turn is the identity on the boundary circle, this map is the identity near both boundary components of the annulus and extends by the identity outside it. It preserves <orientation> and turns one end relative to the other by one full turn. An inverse twist uses $-\rho$.
More intrinsically, $h$ is a <Dehn twist> if its <mapping class> is that of this annulus-supported <homeomorphism>, for some curve $c$ and the chosen handedness. Choices of annular coordinates and interpolation give isotopic maps. The genus hypothesis permits essential curves, but the definition also makes sense for boundary-parallel curves on surfaces with boundary.
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