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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