most recent 30 from http://mathoverflow.net 2013-05-22T01:39:53Z http://mathoverflow.net/feeds/question/14530 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/14530/4-genus-of-a-2-bridge-link Megan 2010-02-07T18:51:43Z 2011-06-20T16:48:36Z

<p>How can we calculate the 4-genus of a link L? The 4-genus is defined to be the minimal genus of orientable surface bounded by L in B^4. Is there any routine method to calculate that?</p> <p>Especially, any good idea how to calculate the 4-genus of a 2-bridge link? Thanks.</p>

http://mathoverflow.net/questions/14530/4-genus-of-a-2-bridge-link/14534#14534 Ryan Budney 2010-02-07T19:02:33Z 2010-02-07T19:02:33Z

<p>Take a look at Paolo Lisca's papers "Rational balls and the ribbon conjecture" and "Sums of lens spaces bounding rational balls". He determines which 2-bridge knots are slice and the concordance order for all 2-bridge knots. </p> <p>But no there's no known effective procedure to compute 4-genus.</p>

http://mathoverflow.net/questions/14530/4-genus-of-a-2-bridge-link/14544#14544 Ben Webster 2010-02-07T20:12:54Z 2010-02-07T20:12:54Z

<p>You might get interesting lower bounds using <a href="http://arxiv.org/abs/math.GT/0402131" rel="nofollow">Rasmussen's s-invariant</a>, and his calculations of <a href="http://arxiv.org/abs/math.GT/0508510" rel="nofollow">the KR homology of 2-bridge links</a>. </p> <p>Or you might not. As Ryan says, there are various ways of bounding genus above and below, but unless you get lucky and two of them match, it can be hard to tell. The 4-genus of torus knots was an open problem for a scandalously long time (if my history is right, it was first proven in the 90's!).</p>

http://mathoverflow.net/questions/14530/4-genus-of-a-2-bridge-link/14871#14871 Andrew Lobb 2010-02-10T07:07:50Z 2010-02-10T07:07:50Z

<p>Rasmussen and Lee's results say that the $s$ invariant of a 2-bridge knot will be just equal to the signature of the knot. So you can compute the signature of your knot to get a lower bound (there are extremely rapid ways of doing this from an alternating diagram). Unfortunately the only decent way to get an upper bound that I know of is by spotting a smooth surface! Good luck.</p> <p>Remember that $s$ might not be the best you can do. In particular, among alternating knots, the figure 8 knot ($4_1$ in Rolfsen) has vanishing $s$ invariant (for example, because it is torsion in the concordance group) and yet it is not even slice if you allow your surfaces to be locally flat, let alone smooth.</p>

http://mathoverflow.net/questions/14530/4-genus-of-a-2-bridge-link/68299#68299 Danny Ruberman 2011-06-20T16:38:47Z 2011-06-20T16:48:36Z

<p>(This is really a comment on the answer relating to concordance order.) </p> <p>Since your p is even, then your 2-bridge knot is actually a link. So, while it makes sense to ask if it's a slice or ribbon link, asking about its concordance order doesn't make sense. </p>