A 3-dimensional rendering of the ribbon knot , showing the ribbon property
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 slice disc M is a smoothly embedded in with . Consider the function given by . By a small isotopy of M one can ensure that f restricts to a Morse function on M. One says is a ribbon knot if has no interior local maxima.
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?
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, MR0140100. Reprinted by Dover Books, 2010.