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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.

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A proof of what? –  HJRW Jun 13 '12 at 13:12

3 Answers 3

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.

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Thanks a lot! A will check this. –  Andrew Jun 13 '12 at 13:15

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

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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.

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Here's a simple such algorithm, assuming that your multicurve is expressed as an integer system of weights on a train track: start splitting the train track. Split, split, split. Eventually it falls into integer weighted simple closed curves. Verify whether the result is one curve with weight $1$. –  Lee Mosher Jun 13 '12 at 13:49
This in some sense is a direct analogue of the torus situation: given a rational number expressed as $p/q$, is $gcd(p,q)=1$? To decide, apply the Euclidean algorithm: divide, divide, divide... –  Lee Mosher Jun 13 '12 at 13:52
@Lee: I would love to know more about this (any references you can recommend would be most welcome), but my fear is that the algorithm you get is polynomial in the WEIGHTS, so exponential in the size of the input... But of course, i speak from ignorance. –  Igor Rivin Jun 13 '12 at 14:34
There is a polynomial algorithm in log of the weights in this paper (due to Thurston): ams.org/journals/tran/2006-358-09/S0002-9947-05-03919-X/… –  Ian Agol Jun 13 '12 at 14:52
@LeeMosher - The phrase "split, split. split" in train-track-land actually translates to "subtract, subtract, subtract" in $\gcd$-land. The way that [AHT] and [EN] speed up the algorithm is by (vaguely put) figuring out how to "divide, divide, divide". –  Sam Nead May 10 at 18:40

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