heegard diagram - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:14:28Z http://mathoverflow.net/feeds/question/109261 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109261/heegard-diagram heegard diagram mark 2012-10-09T23:50:02Z 2012-10-11T14:30:21Z <p>It seems like there is an algorithm to find the Heegard diagram of a 3 manifold obtained by surgery on a link. Also someone told me I can find it in the Gompf and Stipciz's book. But I could not find it. Can anyone help? </p> http://mathoverflow.net/questions/109261/heegard-diagram/109295#109295 Answer by Daniel Moskovich for heegard diagram Daniel Moskovich 2012-10-10T13:04:44Z 2012-10-11T08:30:48Z <p>I don't know about Gompf and Stipsicz, but the algorithm to obtain a Heegaard diagram from a surgery presentation may be found for example in <a href="http://www.jstor.org/discover/10.2307/2154000?uid=3738992&amp;uid=2129&amp;uid=2&amp;uid=70&amp;uid=4&amp;sid=21101302227937" rel="nofollow">A Simple Proof of the Fundamental Theorem of Kirby Calculus on Links</a> by Ning Lu, Trans. Amer. Math. Soc. Vol 331(1) pp. 143-156 (1992). It's not difficult, and it's worth knowing.</p> http://mathoverflow.net/questions/109261/heegard-diagram/109307#109307 Answer by Kevin Walker for heegard diagram Kevin Walker 2012-10-10T16:05:40Z 2012-10-11T14:30:21Z <p>(1) Choose a planar presentation of your link, approximately in the plane of the blackboard. Let $N$ be a tubular neighborhood of the link in this position.</p> <p>(2) For each crossing $c$ of the presentation, add a 1-handle $T_c$ to $N$ which is perpendicular to the blackboard and connects the upper and lower parts of the crossing. Let $H$ be the union of $N$ and all the 1-handles $T_c$.</p> <p>(3) $S^3 \setminus H$ is a handlebody. The corresponding set of Heegaard curves on $\partial H$ bijects with the complementary regions of the (flattened) planar diagram of the link.</p> <p>(4) $H$ is, of course, also a handlebody. For the corresponding Heegaard curves take the surgery curves on $N$ (this involves a choice to make them disjoint from the attaching disks of the $T_c$) union the obvious small, disk-bounding curves on the boundary of each $T_c$. (Correction added later: Actually, if there are $k$ components of the link then one should omit $k-1$ of the $T_c$ curves. The omitted crossings should be minimal with respect to connecting the components of the link. Also, I'm assuming that the planar projection is a connected graph.)</p>