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). 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, 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 e i θ {displaystyle e^{i heta }} with θ ∈ [ 0 , 2 π ] , {displaystyle heta in ,} and t ∈ . 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 γ a {displaystyle gamma _{a}} . Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve γ a {displaystyle gamma _{a}} will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say