 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.

A Dehn twist is a fundamental concept in the field of topology, particularly in the study of surfaces and 3-manifolds. It is a type of homeomorphism that can be used to analyze the properties of surfaces and their mappings.