page_title: Dehn twist
section_title: Definition
hierarchical_section_title: Dehn twist / Definition
caption_attribution_description: English: General Dehn twist on a compact surface represented by a n-gon.
context_page_description: In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface.
context_section_description: 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, homeomorphic to the Cartesian product of a circle and a unit interval I: 
  Give A coordinates (s, t) where s is a complex number of the form  with  and t ∈ [0, 1].  
  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.
caption: General Dehn twist on a compact surface represented by a n-gon.
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