Definition
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.
Read more about this topic: Dehn Twist
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