CHAPTER 1

Diagrams of Knotted Surfaces

The classical notion of a knot diagram is reviewed. We discuss the analogous

notion of a knotted surface diagram and give examples of surfaces in 4-space and

their diagrams. A diagram consists of a surface in general position in 3-space

together with crossing information given. We show how to obtain an embedded

surface in 4-space from a diagram, and we show how to obtain a diagram from

a given embedding. We present a combinatorial description of knotted surfaces,

and we review other diagrammatic methods. The main result of this chapter is

Theorem 1.10 in which we give several alternative descriptions of knotted surfaces.

We work in smooth category in this book unless otherwise stated.

1.1. Classical knot diagrams

A knot is a codimension 2 embedding of a closed manifold K : M n _ 2 — Nn.

Usually N is an Euclidean space and M is a sphere. A classical knot is an embedding

of a circle in 3-space. An embedding of more than one circle in 3-space is called a

link.

In the classical case, a knot diagram consists of an immersed closed curve in

the plane that has crossing information indicated at its double points. Let us

explain this definition. First, without loss of generality we may assume that a

planar immersion of a curve is generic in the sense that there are no multiple

points of multiplicity greater than two, and the intersection of arcs of the curve

are transverse (non-tangential). Second, the crossing information is indicated by

deleting a small open neighborhood of the double point on one of the intersecting

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Side View

FIGURE

1. Classical knot diagrams

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http://dx.doi.org/10.1090/surv/055/01