Dehn twist

A Dehn twist about a simple closed curve $\gamma$ in an oriented surface is a homeomorphism supported in an annulus around $\gamma$. In the coordinates and orientation convention of a right-handed Dehn twist, it has the formwhere $f$ is zero near one boundary circle and one near the other. It extends by the identity outside the annulus; subtracting $2\pi f(r)$ gives its inverse. Changing the interpolation without changing the end values gives the same mapping class.

For essential simple closed curves $\alpha ,\gamma$ and every integer $n$,Thus $i(\alpha ,\gamma )>0$ forces all iterates to have distinct isotopy classes. In minimal position of curves or arcs, each crossing strand winds $n$ times through the other strands; the bigon criterion proves the counted intersections cannot be removed.^[ref]

On a torus, take curves representing a basis $a,b$ of first homology, with $a\cdot b=1$. Under the stated shear convention, $T_a^n(b)=b+na$, and its geometric intersection number with $b$ is $|n|$. A twist about a curve bounding an unpunctured disc is instead isotopic to the identity by the Alexander trick. Boundary-parallel twists can remain nontrivial when isotopies must fix the boundary pointwise.

The mapping class group of a compact connected oriented topological surface, for orientation-preserving maps fixing its boundary pointwise and isotopies relative to that boundary, is generated by Dehn twists about simple closed curves. Boundary-parallel twists are allowed. For a closed surface of positive genus a finite generating collection exists. Permuting boundary components requires additional generators if that larger group is used.

If $\gamma \cdot u$ uses the ordered-tangent orientation convention for algebraic intersection number of curves on an oriented surface, a right-handed Dehn twist takes $u$ to $u+([\gamma ]\cdot u)[\gamma ]$ on first homology. In particular a separating-curve twist acts trivially there.

In orientation-preserving annular coordinates $(\theta ,r)$ with orientation $d\theta \wedge dr$, the twist is $(\theta ,r)\mapsto (\theta +2\pi f(r),r)$, where $f$ increases from zero to one. The map is the identity outside this annulus and near its boundary. The orientation of the ambient topological surface, rather than the orientation assigned to the curve, fixes the handedness.