A knot is called a ribbon knot if it bounds a ribbon disk; that is, an immersed disk in three-space with singular set a disjoint collection of double arcs, each one of which has preimage on the two-disk consisting of a pair of arcs, one with boundary on the circle, and the other interior to the disk. The Ribbon Number is the minimum number of such double arcs among all ribbon disks. Early references include [3,4].
If K is not a ribbon knot, we note that the ribbon number does not exist: "D.N.E."
If K is ribbon, then it bounds an embedded ribbon disk in the four-ball. (All known slice knots among prime knots with 13 or fewer crossings are known to be ribbon knots.) The Fusion Number of a knot K is the minimum number of saddle points among all embedded ribbon disks. This equals the minimum number of bands in a disk-band presentation of K in three space. For all K, ribbon_number(K) >= fusion number(K) +1.
Thanks go to Alex Zupan for providing us with the results of [1, 2] and for assisting us in preparing the data to include in KnotInfo.
[1] Xianhao An, Matthew Aronin, Aaron Banse, David Cates, Ansel Goh, Benjamin Kirn, Josh Krienke, Minyi Liang, Samuel Lowery, Ege Malkoc, Jeffrey Meier, Max Natonson, Veljko Radic, Yavuz Rodoplu, Bhaswati Saha, Evan Scott, Roman Simkins, and Alexander Zupan. Ribbon numbers of 12-crossing knots, Arxiv preprint.
[2] Stefan Friedl, Filip Misev, and Alexander Zupan. Bounding the ribbon numbers of knots and links. Arxiv preprint.
{3] Yoko Mizuma, An estimate of the ribbon number by the Jones polynomial, Osaka J. Math. 43 (2006), no. 2, 365-369.
[4] Yoko Mizuma and Yukihiro Tsutsumi, Crosscap number, ribbon number and essential tangle decompositions of knots, Osaka J. Math. 45 (2008), no. 2, 391-401.