Timeline for heegard diagram
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 9, 2014 at 19:11 | comment | added | Kevin Walker | In the lens space example, the two curves of the Heegaard diagram are (1) a $-p/q$ curve on $\partial N$, and (2) a curve which bounds a disk in $S^3\setminus N$, i.e. a longitude. | |
| Sep 9, 2014 at 16:31 | comment | added | Manuel Bärenz | How does this work for example for a Lens space $L(p,q)$? That's just $-p/q$-surgery on the unknot. (1) So we take $N = S^1 \times D^2 \stackrel{\text{unknot }-p/q}{\hookrightarrow} S^3$. (2) We don't need any 1-handles since there are no crossings. $H = N$. (3) I can only guess what you mean by this step. I'm drawing the circle ${0} \times \partial D^2 \subset \partial H$. (4) What exactly are the surgery curves on $N$? In particular, for general $p,q$, the curve must go around $H$ several times? I take it this is just the torus knot? | |
| Oct 11, 2012 at 14:30 | history | edited | Kevin Walker | CC BY-SA 3.0 |
small correction
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| Oct 11, 2012 at 14:18 | comment | added | Kevin Walker | ... Put another way, the output is a surface with two sets of curves, not two handlebodies with a homeomorphism between their boundaries. | |
| Oct 11, 2012 at 14:13 | comment | added | Kevin Walker | I agree that the outline is rough, but on the other hand the question asked for an algorithm, not a proof that the algorithm was correct. ... Actually, I don't understand your comment. I don't think I ever have to mention or think about Dehn twists, and I think what I wrote works fine for non-unit and even non-integer surgeries. | |
| Oct 11, 2012 at 8:32 | comment | added | Daniel Moskovich | This is a nice very-rough-outline! There's the "digging a trench" construction needed to prove that surgery along the curve gives rise to Dehn twists on the handlebody; you also have to do something non-trivial to deal with non-unit framings. | |
| Oct 10, 2012 at 16:38 | vote | accept | mark | ||
| Oct 10, 2012 at 16:05 | history | answered | Kevin Walker | CC BY-SA 3.0 |