Generating ribbon diagrams for knots known to be ribbon knots - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T03:31:56Z http://mathoverflow.net/feeds/question/8723 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/8723/generating-ribbon-diagrams-for-knots-known-to-be-ribbon-knots Generating ribbon diagrams for knots known to be ribbon knots Ryan Budney 2009-12-13T00:59:41Z 2012-10-15T09:37:50Z <p>Is there a source in the literature for ribbon diagrams for the knot-table knots known to be ribbon knots?</p> <p>For example, I'm interested in doing a computation which needs as input a ribbon diagram for the knot \$8_{20}\$ (Rolfsen knot table notation). This knot is known to be ribbon, but I don't know a ribbon diagram for the knot. </p> <p>Usually when I encounter a claim of the sort "knot X is ribbon" either the author supplies the ribbon diagram, or nothing. Citations to information of this sort seem kind of sparse. Or am I just unaware of a standard source for this type of information? </p> http://mathoverflow.net/questions/8723/generating-ribbon-diagrams-for-knots-known-to-be-ribbon-knots/8728#8728 Answer by Scott Carter for Generating ribbon diagrams for knots known to be ribbon knots Scott Carter 2009-12-13T01:30:55Z 2009-12-13T02:01:15Z <p>I think Kawauchi's book has tables that include ribbon diagrams, but I don't have a copy with me. Look at <a href="http://www.indiana.edu/~knotinfo/diagrams/8_20.png" rel="nofollow"> Livingston and Cha </a>. It is not hard to get a ribbon disk from this diagram: add a handle between the ears on the top and bottom right. </p> <p>Generally, I check <a href="http://www.indiana.edu/~knotinfo/" rel="nofollow"> Livingston/Cha </a>, <a href="http://www.math.toronto.edu/~drorbn/KAtlas/Knots/8.20.html" rel="nofollow"> Bar-Natan, </a> and <a href="http://shell.cas.usf.edu/quandle/Invariants/database/database.php" rel="nofollow"> Saito </a> for various information.</p> <p>@ears: there are a pair of symmetric clasps on the top and bottom of the diagram. Pull the top-most and bottom-most arc to the right, and then attach a band. The vertical arc that forms a triangle, and the right vertical arc from the band forms an obvious embedded circle. </p> http://mathoverflow.net/questions/8723/generating-ribbon-diagrams-for-knots-known-to-be-ribbon-knots/8781#8781 Answer by Scott Morrison for Generating ribbon diagrams for knots known to be ribbon knots Scott Morrison 2009-12-13T18:50:05Z 2009-12-13T18:50:05Z <p>The <a href="http://katlas.org" rel="nofollow">Knot Atlas</a> seems like it would be a good home for ribbon diagrams. At present, it has exactly one: for the knot <a href="http://katlas.org/wiki/6%5F1" rel="nofollow">6<sub>1</sub></a>.</p> <p>The Knot Atlas is a wiki, it's user editable, and you can even upload new images.</p> http://mathoverflow.net/questions/8723/generating-ribbon-diagrams-for-knots-known-to-be-ribbon-knots/8807#8807 Answer by Ryan Budney for Generating ribbon diagrams for knots known to be ribbon knots Ryan Budney 2009-12-13T22:46:32Z 2009-12-13T22:46:32Z <p><img src="http://katlas.org/w/images/0/05/8%5F20.ribbon.png" alt="alt text" /></p> <p>tada: <a href="http://katlas.org/wiki/8%5F20" rel="nofollow">link text</a></p> http://mathoverflow.net/questions/8723/generating-ribbon-diagrams-for-knots-known-to-be-ribbon-knots/109699#109699 Answer by Christoph Lamm for Generating ribbon diagrams for knots known to be ribbon knots Christoph Lamm 2012-10-15T09:37:50Z 2012-10-15T09:37:50Z <p>See also 'A refined Jones polynomial for symmetric unions', Michael Eisermann and Christoph Lamm, Osaka J. Math. (2011), <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ojm/1315318344" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.ojm/1315318344</a></p> <p>All prime ribbon knots up to 10 crossings are given as symmetric diagrams (examples 1.14, 6.6, 6.7 and 6.8, see especially the table on page 363) which are simpler than the diagrams in Kawauchi's book.</p>