In this book the authors develop the theory of knotted
surfaces in analogy with the classical case of knotted curves in
3-dimensional space. In the first chapter knotted surface diagrams
are defined and exemplified; these are generic surfaces in 3-space
with crossing information given. The diagrams are further enhanced to
give alternative descriptions. A knotted surface can be described as a
movie, as a kind of labeled planar graph, or as a sequence of words in
which successive words are related by grammatical changes.
In the second chapter, the theory of Reidemeister moves is
developed in the various contexts. The authors show how to unknot
intricate examples using these moves.
The third chapter reviews the braid theory of knotted
surfaces. Examples of the Alexander isotopy are given, and the braid
movie moves are presented. In the fourth chapter, properties of the
projections of knotted surfaces are studied. Oriented surfaces in
4-space are shown to have planar projections without cusps and without
branch points. Signs of triple points are studied. Applications of
triple-point smoothing that include proofs of triple-point
formulas and a proof of Whitney's congruence on normal Euler
classes are presented.
The fifth chapter indicates how to obtain presentations for the
fundamental group and the Alexander modules. Key examples are worked
in detail. The Seifert algorithm for knotted surfaces is presented and
exemplified. The sixth chapter relates knotted surfaces and
diagrammatic techniques to 2-categories. Solutions to the
Zamolodchikov equations that are diagrammatically obtained are
presented.
The book contains over 200 illustrations that illuminate the
text. Examples are worked out in detail, and readers have the
opportunity to learn first-hand a series of remarkable geometric
techniques.
Readership
Graduate students, research mathematicians, physicists,
and computer graphics experts interested in knots and links.