= Dehn twist
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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 form
$$
(\theta,r)\longmapsto(\theta+2\pi f(r),r),
$$
where $f$ 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> $n$,
$$
i(T_\gamma^n\alpha,\alpha)=|n|\,i(\alpha,\gamma)^2.
$$
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.<A primer on mapping class groups (Version 5.0) by Benson Farb and Dan Margalit>[§3.1 and Proposition3.2, printed pages72–73]{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.
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