10 1. DIAGRAMS OF KNOTTED SURFACES

FIGURE

9. Repairing broken surfaces at a double curve

Now consider the set

C = {(x, y,z,w):y = 0, w 0,

z2+w2

= 1, &

x2

+

z2

+

w2

4}.

The set C forms a trough in (x,z,u)-space that can be attached to the neigh-

borhood of the broken diagram. Upon projection to the w = 0 hyperplane, the

union C U B will project to the disk

x2

+

y2

4 which will intersect T along a seg-

ment of the x-axis. Figure 9 contains an illustration of the repair work. On the left

of the figure we see the broken surface in (x,?/,z)-space (w = 0). On the right, we

illustrate the piece of the surface that dips below this hyperplane, by drawing the

surface in (x,z, w)-space (y = 0). Observe that the arc along the x-axis indicates

the intersection between the (x,?/)-plane and the (x,z, w)-space.

At a triple point, we perform a similar, but slightly more complicated, repair.

Assume that the surface in the neighborhood, N(a), consists of the following sets.

T = {(x,y,z,w) : x2 + y2 + z2 4, w = 0, & z = 0}, M = {(x,y,z,w) :

x2 + y2 + z2 4, w = 0, 2/ = 0, & 1 |z|}, and £ = {(x, j /, z, w) : x2 -f y2 H- z2

4, w = 0, x = 0, 1 \y\, & 1 \z\} (T,M, and 5 stand for top, middle, and

bottom). Then in the (x, z, it;)-space y = 0 we can repair the sheet M by attaching

a trough of the form {(x, 0, z, w) : w 0,

z2

-f

w2

= 1, k

x2

+

z2

+

w2

4} (See

the lower left illustration of Fig. 10). We repair the bottom sheet by attaching a

cross trough in the (y,z, w;)-space x — 0. The shape of the trough is indicated on

the right of Fig. 10. On the boundaries of the trough are semi-circles of radius 1,

but the regions interior is deep enough so that the repair of the middle sheet fits

into the trough that repairs the bottom sheet.

The illustration in Fig. 11 indicates how to fill in the surface in a neighborhood

of a branch point.

Observe that the construction of an embedded surface from a diagram is an in-

verse operation to constructing a diagram from an embedded surface. Furthermore,

there are many diagrams that can be associated to a given embedding. In Chapter

2, we discuss the theory of moves to knotted surface diagrams that is analogous

to the classical Reidemeister theory. First, we remind the reader of the classical

theory so that we can discuss some of the intricacies of knotted surface diagrams.