 Proof of the Tangle Characterization of Ribbon Knots
Theorem. A knot K is ribbon iff there exists a tangle T whose 𝜏 closure is the untangle and whose κ closure is K.
Proof. The backward ⟸ implication is easy:
For the forward implication, follow the following 5 steps:
Step I: In-situ cosmetics.
At end: D is a tree of chord-and-arc polygons.
Step 2: Near-situ cosmetics.
At end: D is tree-band-sum of n unknotted disks.
Step 3: Slides.
At end: D is a linear-band-sum of n unknotted disks.
Step 4: Exposure!
The green domain is contractible - so it can be shrank,
moved at will (with the blue membrane following along),
and expanded back again.
At end: D has (n-1) exposed bridges which when turned,
make D a union of n unknotted disks.
Step 5: Pulling bottom handles
avoiding the obstacles.
At end: Theorem is proven.