2

1. DIAGRAMS OF KNOTTED SURFACES

arcs. By convention, the arc that has a segment deleted is thought to be farther

away from the plane of projection than the unbroken arc is.

The generic projection of a given embedding K : S

1

-• R

3

onto a plane in R

3

engenders a diagram because the arc of the knot K that is farther away from that

plane at a double point of the projection is eclipsed by the arc that is closer to the

plane. Thus a given knot has a diagram.

On the other hand given a diagram, one can construct a knot as follows. Con-

sider the family of embedded arcs in the plane that constitute the diagram. Then

at each crossing point, the broken arc is repaired by stringing an arc in 3-space

below the plane between the two broken ends at the would-be double point. An

embedded closed curve results that will project to the knotting depicted by the

diagram. Figure 1 illustrates the technique.

1.2. Knotted surface diagrams

The main concern of this book is knots in 4-space. We call an embedding

K : F — R

4

of a closed surface F (compact without boundary) a knotted surface,

or simply a knotting. It is also sometimes said that an embedding F C R 4 is a

knotted surface; that is we consider the image K(F) and suppress the map K in the

notation. In fact, by definition, a knotted surface may be "unknotted." A sphere

is called unknotted if it bounds an embedded 3-ball in 4-space. However we use the

term knotted surface to represent any embedding. Before we give the definition of

a knotted surface diagram that is analogous to the classical case, we establish some

preliminary definitions.

Generic surfaces in R

3

. Let F denote a closed surface (compact without

boundary). We say that a smooth map / : F -» R

3

denotes a generic surface

if for each point x G F there is a 3-ball neighborhood N(y) containing y = f{x)

such that the pair (N(y),f(F) C\ N(y)) is diffeomorphic to (£?3, the intersection

of i coordinate planes) (i = 1,2,3, B3 is a 3-ball containing the origin), or (B3,

the cone on a figure 8) where the figure 8 curve (a lemniscate) is in the boundary

of N(y). In the second case the surface can be parametrized as the image of the

surface (x,y) •-

(x2,y,xy).

The set

5

i

= { y € R

3

: # / -

1

( ! / ) = i }

is called the j-tuple point set. In case j = 2,3, then this is called the double

point set or the triple point set, respectively. If y G R 3 is a point such that for

any neighborhood N(y) the intersection of N(y) and the image f(F) contains a

cone on a figure 8, then y is said to be a branch point, pinch point or Whitney's

umbrella of the surface [GG]. Observe that the branch points and the triple points

are in the closure of the double point set. The left three pictures in Fig. 2 show

neighborhoods of a double point, a triple point, and a branch point from top to

bottom, respectively. The term generic is used to describe such surfaces because

the set of such maps form an open dense subset in the space of all smooth (= C°°)

maps. In particular, given any smooth map from F to 3-space, it is possible to

perturb the map slightly into one that is generic.

The closure of the double point set is the image of a compact 1-dimensional

manifold immersed (but not generically so) in 3-space. The most elegant way of