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.

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 ${\displaystyle f:D^{4}\to \mathbb {R} }$ 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 ${\displaystyle f_{|M}:M\to \mathbb {R} }$ has no interior local maxima.

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