# A simple closed curve on a surface

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

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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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