Ribbon disks and ribbon knots

Let $D^2$ be the oriented closed disk and let $K\subset S^3=\mathrm{\partial }{B}^4$ be a smooth oriented knot, meaning the image of an orientation-preserving smooth embedding of a circle. A ribbon immersion for $K$ is a smooth immersion $f:D^2\to S^3$ whose boundary restriction is an embedding onto $K$ and whose only multiple points form finitely many pairwise disjoint transverse double arcs ${\gamma }_1,\dots ,{\gamma }_m$. For each $i$, the full preimage of ${\gamma }_i$ consists of two disjoint embedded arcs ${\alpha }_i$ and ${\beta }_i$ in $D^2$: ${\alpha }_i$ is properly embedded, with its two endpoints on $\mathrm{\partial }{D}^2$ and its interior in $\mathrm{Int}{D}^2$, while ${\beta }_i$ is compact and lies in $\mathrm{Int}{D}^2$. There are no triple points or other self-intersections. A ribbon disk for $K$ is such a ribbon immersion, and $K$ is a ribbon knot if it admits a ribbon disk.

A band on a link in $S^3$ is the image of a smoothly embedded rectangle whose two opposite attaching edges lie on the link and whose remaining points are disjoint from it. A band move removes the attaching edges, inserts the other two edges, and rounds the corners; the embedded band, including its half-twist data, is part of the move. A no-maximum disk movie is the family of regular level links of a smooth proper disk $D\subset B^4$ whose boundary collar in $D$ is parametrized by $c_D(y,s)=(1-s)y$ for $y\in \mathrm{\partial }D\subset S^3$ and $0\le s<\epsilon$ for some $\epsilon >0$, and for which the squared radial function $\rho (x)=\parallel x{\parallel }^2$ is Morse in the disk interior with only index-$0$ minima and index-$1$ saddles. A minimum creates a split unknot and a saddle performs an oriented band move; there are no maxima. The no-maximum disk movie is a second presentation convention used by the design for explicit ribbon examples. This movie model and the ribbon-immersion definition above describe the same class of knots: a finite birth-and-band induction converts the movie to an immersion, and cutting the proper double-arc preimages converts an immersion to a disk-and-band movie. The local resolution lemma proves only that a ribbon immersion pushes rel boundary to a smooth proper disk in $B^4$. The slice-ribbon conjecture asks whether every smooth slice knot admits a ribbon immersion as defined above.

Proper smooth surfaces and boundary collars

Smooth oriented knot convention

Smooth sliceness and the ACω slice-genus convention

The Axiom of Countable Choice used by the slice-genus convention

The proved conversion between the two disk presentations ↗

Ray, Slice Knots and Knot Concordance, §1.1

Owens, Knots and 4-Manifolds, Lecture 1

Livingston, A Survey of Classical Knot Concordance, §2.1

Brejevs, ribbon band-move convention, §2.2

The two shaded rectangles are local image sheets in coordinates $(x,y,z)$. They meet transversely along the red double arc $\gamma =\{(x,0,0):-1\le x\le 1\}$. The gray patch ends on the boundary knot $K$ at the two endpoints of $\gamma$; the blue patch is cropped for display but extends past both endpoints, so they lie in the interior of that sheet. Their source preimages are respectively the proper arc $\alpha$ and the interior arc $\beta$. The diagram shows only this local type; it does not encode a global knot diagram or serve as a proof.

The plan retains ${\mathrm{A}\mathrm{C}}_{\omega }$ because the library's smooth slice-genus definition assumes it. The finite definition of a ribbon immersion and its local height separation use no choice principle. The separate presentation-equivalence lemma proves both conversions; the local resolution lemma alone proves only the rel-boundary embedded resolution.