Question

How to describe a simple closed curve on an oriented surface of genus g? I know the answer only for the torus. It would be nice to find an article or a book where proof can be found.

Answer 1

There are multiple ways, depending on what your goal is.

From an algorithmic point of view, if you are given a triangulation of the surface, normal curves are a very efficient way to describe a simple closed curve. You can check the recent paper by Jeff Erickson and Amir Nayyeri : http://compgeom.cs.uiuc.edu/~jeffe/pubs/tracing.html for the background.

If you want to describe curves up to isotopy, you can use Dehn-Thurston coordinates. Quick googling gave me the following link : http://intlpress.com/CAG/2004/12-1/CAG_12_1_1_41.pdf but there are probably better references.

Finally, train tracks have proved to be a very useful method of describing curves, especially in the setting of mapping class groups. There is an orange book by Penner and Harer on them, Combinatorics of Train Tracks if I recall correctly. There is also a great survey by Lee Mosher in the Notices of the AMS.

Answer 2

Check out the book Thurston's Work on Surfaces for a treatment of Dehn-Thurston coordinates which is simultaneously intuitive and in-depth.

Answer 3

First, a remark on the other two answers: The Dehn-Thurston coordinates describe multicurves (that is, disjoint unions of essential simple curves). Figuring out when a multicurve is connected (so, an actual curve) is a very difficult computational problem, though it is know through the work of yours truly (Simple curves on surfaces) and Mirzakhani (I suggest taking a look at my paper "A simpler proof of Mirzakhani's simple curve asymptotics") that there is a positive probability (which Mirzkhani expresses in terms of volumes of moduli spaces) that a multicurve (given by D-T coordinates) is a curve.

Given an element in the fundamental group, there are algorithms (Birman-Series, M. Cohen-Lustig, M. Lustig for closed surfaces) to determine whether this element represents a simple closed curve -- unfortunately, this is a decision procedure, and not a method to generate all simple closed curves.