Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).

Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus and so is homeomorphic to the Cartesian product of

where I is the unit interval. Give A coordinates (s, t) where s is a complex number of the form

with

and t in the unit interval.

Let f be the map from S to itself which is the identity outside of A and inside A we have

Then f is a Dehn twist about the curve c.

Dehn twists can also be defined on a non-orientable surface S, provided one starts with a 2-sided simple closed curve c on S.

Consider the torus represented by a fundamental polygon with edges a and b

Let a closed curve be the line along the edge a called ${\displaystyle \gamma _{a}}$.

Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve ${\displaystyle \gamma _{a}}$ will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say

...
Wikipedia