Dehn twist on an oriented surface

Let $F$ be an oriented smooth surface and let $c:S^1\hookrightarrow F$ be a smooth embedded circle. Choose an annular coordinate map $\varphi :S^1\times [-1,1]\to A\subset F$ whose product orientation agrees with the orientation of $F$ and whose core $\varphi (S^1\times \{0\})$ is $c(S^1)$. Choose a smooth function $\rho :[-1,1]\to [0,1]$ that is $0$ near $-1$ and $1$ near $1$. The positive Dehn twist about $c$ in the chosen annular coordinates is the diffeomorphism that is the identity outside $A$ and on $A$ is

$$\varphi (e^{i\theta },t)⟼\varphi (e^{i(\theta +2\pi \rho (t))},t)\mathrm{.}$$

The negative Dehn twist is its inverse. The annular coordinates and cutoff are part of the data whenever a representative diffeomorphism is used; its mapping class is its class under smooth isotopy through orientation-preserving diffeomorphisms of $F$.

Orientation of the surface

The embedded circle

Kirby, A Calculus for Framed Links in S3, §2, printed pp. 36–37