# Generating ribbon diagrams for knots known to be ribbon knots

Is there a source in the literature for ribbon diagrams for the knot-table knots known to be ribbon knots?

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.

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?

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## 4 Answers

I think Kawauchi's book has tables that include ribbon diagrams, but I don't have a copy with me. Look at Livingston and Cha . It is not hard to get a ribbon disk from this diagram: add a handle between the ears on the top and bottom right.

Generally, I check Livingston/Cha , Bar-Natan, and Saito for various information.

@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.

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I have Kawauchi's book in my office. Maybe I missed the ribbon diagrams? "add a handle between the ears..." I'm not so sure what this means. –  Ryan Budney Dec 13 '09 at 1:40
Oh, thanks. Yes, it's in Kawauchi. Google books indexes the ribbon diagrams, too! Merry Christmas from Google books, saves me a trip to the office. –  Ryan Budney Dec 13 '09 at 1:53

tada: link text

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Very pretty indeed! –  Sam Nead Dec 13 '09 at 23:31

See also 'A refined Jones polynomial for symmetric unions', Michael Eisermann and Christoph Lamm, Osaka J. Math. (2011), http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ojm/1315318344

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.

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The Knot Atlas seems like it would be a good home for ribbon diagrams. At present, it has exactly one: for the knot 61.

The Knot Atlas is a wiki, it's user editable, and you can even upload new images.

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I'd be happy to do that but editing that wiki appears very complicated as everything is auto-generated -- using the "edit" link would likely mean my edit would be auto-erased the next time the page is generated? –  Ryan Budney Dec 13 '09 at 21:05
It is a bit evil. The right thing to do is to find the link [edit Notes for 6_1's three dimensional invariants]. That takes you to a sub-page which is not automatically generated, but which is 'transcluded' into the main page for 6_1. –  Scott Morrison Dec 13 '09 at 22:20
Okidokie, I think I may have done the right thing. ? –  Ryan Budney Dec 13 '09 at 22:38