Ribbon knot

A square knot drawn as a 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 formally, this type of singularity is a self-intersection along an arc; 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.

Morse-theoretic formulation

A slice disc M is a smoothly embedded $D^{2}$ in $D^{4}$ with $M\cap \partial D^{4}=\partial M\subset S^{3}$. Consider the function $f:D^{4}\to {\mathbb R}$ given by $f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}$. By a small isotopy of M one can ensure that f restricts to a Morse function on M. One says $\partial M\subset \partial D^{4}=S^{3}$ is a ribbon knot if $f_{{|M}}:M\to {\mathbb R}$ 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 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-strand pretzel knots. 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.

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, N.J.: 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, doi:10.2140/gt.2010.14.2305, MR 2740649.
• 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.
• Kauffman, Louis H. (1987), On Knots, Annals of Mathematics Studies, 115, Princeton, NJ: Princeton University Press, ISBN 0-691-08434-3, MR 907872.
• Lisca, Paolo (2007), "Lens spaces, rational balls and the ribbon conjecture", Geometry & Topology, 11: 429–472, doi:10.2140/gt.2007.11.429, MR 2302495.