page_title: Dehn twist
section_title: Example
hierarchical_section_title: Dehn twist / Example
caption_attribution_description: English: An example of a Dehn twist on the torus, along the closed curve a, in blue. Where a is an edge of the fundamental polygon representing the torus.
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: 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 .
  Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve  will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say 
  in the complex plane.
  By extending to the torus the twisting map  of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of , yields a Dehn twist of the torus by a.
  This self homeomorphism acts on the closed curve along b. In the tubular neighborhood it takes the curve of b once along the curve of a.
  A homeomorphism between topological spaces induces a natural isomorphism between their fundamental groups. Therefore one has an automorphism
  where [x] are the homotopy classes of the closed curve x in the torus. Notice  and , where  is the path travelled around b then a.
caption: An example of a Dehn twist on the torus, along the closed curve a, in blue, where a is an edge of the fundamental polygon representing the torus.
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