Is the Gompf–Scharlemann–Thompson knot ribbon?

Fox's slice–ribbon conjecture (1962) says that every knot bounding a smooth disk in the 4-ball bounds one of a special, combinatorial kind. Gompf, Scharlemann and Thompson found slice knots that are not known to be ribbon, and these are the leading candidate counterexamples. The task is to show that one of them is ribbon after all, by writing down an explicit ribbon disk, or to prove that it is not.

Problem

A knot $K \subset S^3$ is slice if it bounds a smoothly embedded disk in the 4-ball $B^4$, and ribbon if it bounds such a disk with no local maxima of the radial function; combinatorially, $K$ is ribbon exactly when some $b$ band moves turn it into an unlink of $b + 1$ components. Every ribbon knot is slice.

Task. For the 48-crossing diagram of a GST knot published by Epoch (PD code below), give a sequence of Reidemeister moves and $b$ ribbon band moves that ends in a crossingless unlink with $b + 1$ components. Such a sequence is an explicit ribbon disk. Alternatively, prove that this knot is not ribbon, which would refute the slice–ribbon conjecture.

[[83, 94, 84, 95], [92, 81, 93, 82], [72, 95, 73, 96], [73, 84, 74, 85],
 [93, 70, 94, 71], [82, 71, 83, 72], [2, 10, 3, 9], [17, 5, 18, 4],
 [7, 19, 8, 18], [8, 23, 9, 24], [3, 25, 4, 24], [31, 1, 32, 96],
 [27, 12, 28, 13], [38, 5, 39, 6], [6, 37, 7, 38], [1, 46, 2, 47],
 [28, 44, 29, 43], [47, 33, 48, 32], [51, 21, 52, 20], [21, 53, 22, 52],
 [34, 53, 35, 54], [50, 35, 51, 36], [54, 50, 55, 49], [58, 20, 59, 19],
 [22, 60, 23, 59], [33, 60, 34, 61], [57, 36, 58, 37], [61, 49, 62, 48],
 [55, 63, 56, 62], [63, 57, 64, 56], [13, 69, 14, 68], [42, 69, 43, 70],
 [29, 75, 30, 74], [76, 12, 77, 11], [15, 79, 16, 78], [77, 27, 78, 26],
 [40, 79, 41, 80], [75, 44, 76, 45], [30, 86, 31, 85], [87, 11, 88, 10],
 [16, 90, 17, 89], [88, 26, 89, 25], [39, 90, 40, 91], [86, 45, 87, 46],
 [65, 80, 66, 81], [64, 91, 65, 92], [67, 15, 68, 14], [66, 41, 67, 42]]

Background

Fox posed the slice–ribbon question in 1962. If it holds, the simpler notion of ribbon knots could be used to study the apparently richer class of slice knots. Gompf, Scharlemann and Thompson studied two-component links containing the square knot that surger to $(S^1 \times S^2) \# (S^1 \times S^2)$. Such links, which are probably counterexamples to the generalized Property R conjecture, yield slice knots that are not known to be ribbon [1]. Soon after the paper appeared, a Low Dimensional Topology blog post reported that it had lowered the author's confidence in the conjecture from about 60% to about 5%.

What is known

• The GST knots are slice by construction and not known to be ribbon [1].
• Computer searches for ribbon disks by Gukov, Halverson, Manolescu and Ruehle detect many ribbon knots with up to 70 crossings [2], but Epoch reports that they were inconclusive on this knot.
• The ribbon question also matters for attempts to disprove the smooth four-dimensional Poincaré conjecture [2].
• Smaller candidates keep appearing: in 2026 Oliveira-Smith showed that the 18-crossing knot $18_{\mathrm{nh}00000601}$, which is not known to be ribbon, is slice (arXiv:2603.23717, preprint).

What would count as progress

• A move sequence in which every move (R1, R2, R3 or band move) is given with the diagram it produces, starting from the diagram above and ending in an unlink with one more component than the number of bands. Each step can be replayed and compared up to planar isotopy by software such as SnapPy/Spherogram.
• Epoch scores submissions with its own verifier, which applies every move itself and checks the result exactly.
• A proof that this knot is not ribbon, for instance by a new obstruction, would be an even bigger result.

References

1. R. E. Gompf, M. Scharlemann and A. Thompson, Fibered knots and potential counterexamples to the Property 2R and slice-ribbon conjectures, Geom. Topol. 14 (2010), 2305–2347. doi:10.2140/gt.2010.14.2305 (arXiv:1103.1601)
2. S. Gukov, J. Halverson, C. Manolescu and F. Ruehle, Searching for ribbons with machine learning, 2023. arXiv:2304.09304

Source

Imported from FrontierMath: Open Problems, slice-ribbon, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.