On Codimension Two Ribbon Embeddings

We consider codimension-2 ribbon knottings of circles and 2-spheres. We find that if a given ribbon knot has two ribbon disks, those disks are related by ambient isotopy together with a finite number of local modifications to be described. This allows a complete set of moves to be developed for the representation of ribbon 2-knots by abstract or planar graphs. Similar results hold for classical ribbon knots although the planar graphs in that case are more complex. We also use this result to define new invariants for classical ribbon knots in terms of associated ribbon 2-knots. These results also extend to a restricted category of ribbon links. 1 Overview of Results We show that there is a finite set of local moves, termed ribbon intersection moves, on ribbon disks for classical knots and 2-knots which, together with ambient isotopies, relate any two ribbon disks for the isotopic ribbon knots, answering a question of Nakanishi[7]. The local moves also allow us to construct a method of representing ribbon 2-knots as abstract graphs with labels at their vertices. We may also represent them as planar graphs with labels at the vertices, and we find a finite collection of moves which relate any two such diagrams which represent the same knot. For classical knots, we are able to find a similar presentation, but it is more complex, especially compared to the relatively straightforward theory of knot diagrams and Reidemeister moves. However, the result for classical knots does allow us to construct new invariants in terms of 2-knots. We also speculate on the possibility of extending these results to higher dimensions. Acknowledgements The author would like to thank Adam Sikora and William Menasco for their discussions and advice, and also Xiao-Song Lin for bringing this problem to the author’s attention. 2 Ribbon Disks and RI Moves Throughout this paper we work in the smooth category. For the purpose of this exposition we define an n-knot K to be an embedding of S into S. Two n-knots are considered equivalent if there is an ambient isotopy carrying one embedding to the other. In general one can also consider aspherical manifolds or manifolds with boundary, but we wish to only consider spheres for the moment. An nknot K is defined to be ribbon if there is an immersion φ : D → S such that the following hold: φ|∂D = K (inducing the correct orientation), and wherever φ intersects itself it does so as follows: there exist two D ⊂ D, A1, A2 with ∂Ai ⊂ ∂D , φ|Ai an embedding, and there is a n-disk B ⊂ int(A2) such that φ(φ(A1)) = A1 ∪B. φ(D ) is then called a ribbon disk. The places where the ribbon disk fails to be embedded are therefore n-disks; these are called ribbon intersections. The disks like A1 will be