The mapping class group of an oriented topological manifold Y is the group Mod(Y) of orientation preserving homeomorphisms from Y to itself modulo isotopy. In other words, Mod(Y) is the quotient of the (orientation preserving) homeomorphism group Homeo_+(Y) by the connected component of the identity map. A famous theorem of Dehn asserts that the mapping class groups of an oriented surface Σ of genus g is generated by isotopy classes of Dehn twists (even better, for closed surfaces, Lickorish later showed that it is generated by 3g-1 of them and was able to give explicit generators). What, then, is a Dehn twist? If one has an embedded circle c in Σ, a Dehn twist about c is the homeomorphism from Σ to itself obtained by first identifying a tubular neighborhood A of c in Σ with the standard annulus S^1x[0,1] and using this identification to extend the self-homeomorphism of A given by the "twist map" (z,t)\mapsto(e^{2\pi it}z,t) via the identity outside of A. This is usually depicted in books and papers, by necessity, using a static image showing what happens to the line segment {1}x[0,1] under the twist map and, while this gives a good general sense of what is going on, I've always found it a bit unsatisfying since it is somewhat trickier to visualize just how much the rest of the annulus is being twisted so I decided to try my hand at animating the twist map.

To make this animation, I used the following LaTeX code to create a standalone .pdf document containing each of the individual frames in succession and then converted this into a .gif image using an online .pdf to .gif converter. In future, I would like to understand how to use the animate package for LaTeX to accomplish the same result natively.