On Codimension Two Ribbon Embeddings

TL;DR

This paper studies ribbon n-knots for n≥2, establishing moves and isotopies that relate ribbon disks, proving a bijective correspondence between ribbon knots of different dimensions, and exploring diagrammatic calculus and stable equivalence of band presentations.

Contribution

It introduces a set of moves relating ribbon disks, proves a bijection between ribbon knots of different dimensions, and analyzes stable equivalence of band presentations.

Findings

Any two ribbon disks for isotopic knots are related by finite moves and isotopies.

There is a natural bijective correspondence between ribbon n-knots and ribbon m-knots.

Any two band presentations of the same knot are stably equivalent.

Abstract

We consider ribbon n-knots for n\geq 2. For such knots we define a set of moves on ribbon disks, and show that any two ribbon disks for isotopic knots are related by a finite sequence of such moves and ambient isotopies. Using this we are able to prove that there is a natural geometric correspondence between ribbon n-knots and ribbon m-knots which is bijective. We also explore a diagrammatic calculus for such knots. In addition we show that for such knots, any two band presentations for the same knot are stably equivalent.