A simple closed curve on a surface - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:04:47Z http://mathoverflow.net/feeds/question/99434 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99434/a-simple-closed-curve-on-a-surface A simple closed curve on a surface Andrew 2012-06-13T12:17:33Z 2012-06-13T13:32:23Z <p>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.</p> http://mathoverflow.net/questions/99434/a-simple-closed-curve-on-a-surface/99439#99439 Answer by Arnaud for A simple closed curve on a surface Arnaud 2012-06-13T12:45:02Z 2012-06-13T12:45:02Z <p>There are multiple ways, depending on what your goal is. </p> <p>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 : <a href="http://compgeom.cs.uiuc.edu/~jeffe/pubs/tracing.html" rel="nofollow">http://compgeom.cs.uiuc.edu/~jeffe/pubs/tracing.html</a> for the background.</p> <p>If you want to describe curves up to isotopy, you can use Dehn-Thurston coordinates. Quick googling gave me the following link : <a href="http://intlpress.com/CAG/2004/12-1/CAG_12_1_1_41.pdf" rel="nofollow">http://intlpress.com/CAG/2004/12-1/CAG_12_1_1_41.pdf</a> but there are probably better references.</p> <p>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.</p> http://mathoverflow.net/questions/99434/a-simple-closed-curve-on-a-surface/99441#99441 Answer by Lee Mosher for A simple closed curve on a surface Lee Mosher 2012-06-13T13:12:33Z 2012-06-13T13:12:33Z <p>Check out the book <a href="http://press.princeton.edu/titles/9494.html" rel="nofollow">Thurston's Work on Surfaces</a> for a treatment of Dehn-Thurston coordinates which is simultaneously intuitive and in-depth.</p> http://mathoverflow.net/questions/99434/a-simple-closed-curve-on-a-surface/99443#99443 Answer by Igor Rivin for A simple closed curve on a surface Igor Rivin 2012-06-13T13:32:23Z 2012-06-13T13:32:23Z <p>First, a remark on the other two answers: The Dehn-Thurston coordinates describe <em>multicurves</em> (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.</p> <p>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.</p>