1.2. KNOTTE D SURFACE DIAGRAMS 9

The diagram associated to a given knotting. Let us show how to con-

struct a diagram from a given embedded surface in 4-space.

Let F be a smoothly embedded surface in R4. We choose a unit vector v in

4-space and a 3-dimensional hyperplane P in R4 such that F and P are disjoint

and such that the projection 7r of F in the direction v onto P is a generic surface.

Let g : F - P denote this generic surface (g — K\F)- (Strictly speaking a knotted

surface is defined by a smooth embedding of a surface K : F -

R4,

but here we

just consider the image of / and write F for the image to simplify the notation

below.)

For each double point of g, we can compute the distance in R

4

from P to the

corresponding two points in F\ let us say that x\ and X2 are points in F that

project to the same point in P and that #2 is farther from P than x\ is. Then to

obtain the local broken surface diagram of F in a neighborhood of g{x\) = gfa),

we remove from g(F) the image of a small open neighborhood in F of #2-

For each triple point of #, we perform the analogous construction. Let ^1,^2,

and X3 denote three points in F with common image under the generic map g\

assume that x\ is the closest of the three points in question. The diagram in a

neighborhood of the triple point g{x\) — g(x,2) — #(#3) is obtained by removing

from g(F) the images of small neighborhoods of xz and £3.

For branch points, we can use the same convention. An open neighborhood in

one of the sheets involving a branch point is deleted. See Fig. 2 again.

A diagram associated to the given knotting is obtained by removing such small

open neighborhoods for each double point, triple point, and branch point in the

image of the generic map.

For double decker sets of knotted surface projections, we fix the following con-

vention. The double point curve which lies farther away from (resp. closer to) the

projected space is denoted by solid (resp. dotted) line. Thus the solid line is cut

(by removing its neighborhood) in broken surface diagrams. With this convention,

at a triple point, three sheets have the following decker curves. On the top (resp.

middle, bottom) sheet, there is a crossing between dotted and dotted arcs (resp.

solid and dotted arcs, solid and solid arcs). A solid and dotted arcs share an end

point at a branch point. See Fig. 2. (Figure 1 of Chapter 4, and Fig. 3 of Chapter

5 also illuminate these conventions). We will discuss how to draw decker sets in

Section 1.6.

Constructing embedded surfaces from diagrams. Let us consider a neigh-

borhood of a singular point of a generic surface, and consider a local broken surface

diagram in the same neighborhood. We will construct a local embedding of a surface

in 4-space from the broken surface diagram.

Let g : F -» R 3 be a generic surface and a be a singular point in g(F). If a

is a double point, the procedure is analogous to the classical case. We consider the

broken surface to lie in the boundary of a 4-ball, and we repair the broken surface,

by attaching the Cartesian product of a semi-circle and an interval to the surface.

The semi-circle factor will dip below the boundary of the 4-ball in which the broken

surface sits.

For specificity, we give the reparation in particular coordinates. Suppose N(a)

= {(#, y, z, w) : w = 0, &

x2

+

y2

+

z2

4}. We assume that the broken surface in

N(y) can be identified with the sets T = {(x,y,z,w) : z = 0,w = 0, & x2 +y2 4}

and B = {(a;,y, z, w) : y = 0, w = 0,x2 + z2 4, & \z\ 1}.