Ribbon knot

In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ribbon singularities. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part of the disk through the slit. More precisely, this type of singularity is a closed arc consisting of intersection points of the disk with itself, such that the preimage of this arc consists of two arcs in the disc, one completely in the interior of the disk and the other having its two endpoints on the disk boundary.

A 3-dimensional rendering of the ribbon knot ${\displaystyle 8_{20}}$, showing the ribbon property

Morse-theoretic formulation

A slice disc M is a smoothly embedded ${\displaystyle D^{2}}$ in ${\displaystyle D^{4}}$ with ${\displaystyle M\cap \partial D^{4}=\partial M\subset S^{3}}$. Consider the function ${\displaystyle f\colon D^{4}\to \mathbb {R} }$ given by ${\displaystyle f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}}$. By a small isotopy of M one can ensure that f restricts to a Morse function on M. One says ${\displaystyle \partial M\subset \partial D^{4}=S^{3}}$ is a ribbon knot if ${\displaystyle f_{|M}\colon M\to \mathbb {R} }$ has no interior local maxima.

Slice-ribbon conjecture

Every ribbon knot is known to be a slice knot. A famous open problem, posed by Ralph Fox and known as the slice-ribbon conjecture, asks if the converse is true: is every (smoothly) slice knot ribbon?

Lisca (2007) showed that the conjecture is true for knots of bridge number two. Greene & Jabuka (2011) showed it to be true for three-stranded pretzel knots with odd parameters. However, Gompf, Scharlemann & Thompson (2010) suggested that the conjecture might not be true, and provided a family of knots that could be counterexamples to it. The conjecture was further strengthened when a famous potential counter-example, the (2, 1) cable of the figure-eight knot, was shown to be not slice and thereby not a counterexample.^[1][2]

References

• Fox, R. H. (1962), "Some problems in knot theory", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Englewood Cliffs, New Jersey: Prentice-Hall, pp. 168–176, MR 0140100. Reprinted by Dover Books, 2010.
• Gompf, Robert E.; Scharlemann, Martin; Thompson, Abigail (2010), "Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures", Geometry & Topology, 14 (4): 2305–2347, arXiv:1103.1601, doi:10.2140/gt.2010.14.2305, MR 2740649, S2CID 58915479.
• Greene, Joshua; Jabuka, Stanislav (2011), "The slice-ribbon conjecture for 3-stranded pretzel knots", American Journal of Mathematics, 133 (3): 555–580, arXiv:0706.3398, doi:10.1353/ajm.2011.0022, MR 2808326, S2CID 10279100.
• Kauffman, Louis H. (1987), On Knots, Annals of Mathematics Studies, vol. 115, Princeton, New Jersey: Princeton University Press, ISBN 0-691-08434-3, MR 0907872.
• Lisca, Paolo (2007), "Lens spaces, rational balls and the ribbon conjecture", Geometry & Topology, 11: 429–472, arXiv:math/0701610, doi:10.2140/gt.2007.11.429, MR 2302495, S2CID 15238217.

References

1. Dai, Irving; Kang, Sungkyung; Mallick, Abhishek; Park, JungHwan; Stoffregen, Matthew (28 July 2022). "The $(2,1)$-cable of the figure-eight knot is not smoothly slice". arXiv:2207.14187 [math.GT].
2. Sloman, Leila (2 February 2023). "Mathematicians Eliminate Long-Standing Threat to Knot Conjecture". Quanta Magazine.

External links

• Sloman, Leila (18 May 2022). "How Complex Is a Knot? New Proof Reveals Ranking System That Works". Quanta Magazine.