By I.G. Macdonald

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E X A M P L E S : H O W T O D R A W C H A R T S A N D D E C K E R C U R V E S FIGURE 21 21. A classical ribbon knot FIGURE 22. A ribbon sphere (n -h 2)-space satisfying the above condition (*) where the dimension of D\ and D\ is n. There is an alternative description. Let Sf, • • • , S£ be standardly embedded spheres (unlinked, unknotted in the sense that they bound disjoint balls). Let A{, i = 1, • • • , A; — 1, be embedded disjoint arcs connecting Sf to S™+1. Take disjoint thin tubular neighborhood of Ai and use them as tubes connecting the spheres, yielding a sphere as a result.

When a double point arc crosses a fold line so that, in the projection onto the retinal plane, the double point arc crosses the fold line tangentially, this is called a camel-back move or a ijj-move. In the chart this is a 4-valent vertex with linking circle giving a (solid, solid, dotted, dotted) intersection sequence. 8. When orders of maximal/minimal/crossing points are changed the move is called a locality move. These are depicted in Fig. 20. These are the 4-valent vertices in which a pair of arcs (of either possible color) crosses.

Take disjoint thin tubular neighborhood of Ai and use them as tubes connecting the spheres, yielding a sphere as a result. It can be seen (exercise) that this gives a ribbon knot, and that any ribbon knot has such a description. This construction in dimension 4 is depicted in Fig. 22 where in fact its projection into 3-space is depicted. Its broken surface diagram is depicted in Fig. 23. More generally, in the case n = 1 (resp. n = 2), a ribbon link (resp. a knotted ribbon surface) is denned similarly.