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.
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 local maxima.
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.
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