Dehn Twist - Definition

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.

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