Preface

The purpose of this book is to develop the diagrammatic theory of knotted

surfaces in 4-dimensional space in analogy with the classical theory of knotted and

linked circles in 3-space. This goal may sound unachievable to some readers, how

can we perceive phenomena that occur in 4-space? A related issue is perception in

3-dimensional space which we discuss briefly.

For sighted people, the 3-dimensional world is projected upon a 2-dimensional

surface, the retina, at any particular moment. Spatial relations are determined by

a pair of two-dimensional images. The sense of touch is patently two-dimensional

since the world comes into contact with us by means of our skin — a 2-dimensional

surface. Sound is perceived as vibrations along a 2-dimensional membrane. The

sense of taste is discrete with four states that are either excited or not. Only the

olfactory sense has a degree of multi-dimensionality; it is difficult to classify and

relate various smells. Each odor seems to be an independent quantity. But the

sense of smell may also be discrete with an enormously large set of states.

Since we usually perceive the world by a series of 2-dimensional impression,

how do we come up with a 3-dimensional model of it? One can argue that the

relative position of the wrist, elbow, and shoulder allow for a 3-dimensional world.

So even though the world projects to us on our skin, retinas, or eardrums, we see

the world as 3-dimensional. Additional perceptual clues come from moving the eyes

to see around a corner, moving the hands to feel a different facet, or turning the

ear towards a sound.

So in order to develop the diagrammatic theory of knotted surfaces in 4-

dimensions, we will project the surfaces into 3-dimensions, and we will move the

surfaces around to see their different facets. It is not unreasonable that we will

develop some 4-dimensional intuition in the process.

In classical knot theory, invariants (Alexander, Conway, Jones, HOMPTFLY,

Kauffman polynomials) are computed diagrammatically. Category theoretical in-

terpretations of knot diagrams play a key role in the study of quantum invariants.

The braid form of a classical knot, which is both algebraic and diagrammatic, is

used not only to define new invariants but also as geometric machinery for the

study of knots. Most of these concepts and computations can be generalized to

4-dimensions via diagrams. Thus we will develop the theory of knotted surfaces

and thereby provide the machinery for algebraic and geometric computations.

Here is the outline of the book.

Chapter 1 develops the notion of a knotted surface diagram. A diagram consists

of a generic surface in 3-space together with crossing information indicated along

the double point curves. Such a diagram can be projected further onto a plane

to create a chart — a planar graph with labeled edges. The chart can be used to

reconstruct the surface and to construct two other models. A movie consists of the

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