diagram with a fixed height function in 3-space. In such a movie we can consistently
fix the height functions in the stills to create a fully combinatorial description of
the surface. The combinatorial description is called a sentence; this is a sequence
of words that are related by some grammatical rules. We give examples of each of
these descriptions, and discuss some other diagrammatical methods.
Chapter 2 contains the theory of Reidemeister moves. For each description of
the knotted surfaces there is a finite set of moves such that equivalent knottings
are related by a finite sequence of moves taken from this set. We give examples of
applications of the moves in the various context. Chapter 2 closes with the classical
argument that a coffee cup and a doughnut represent isoptopic surfaces in 3-space.
Chapter 3 reviews the theory of surface braids that has been extensively devel-
oped by S. Kamada. We discuss generalizations of Alexander, Markov and Artin
theorems. In particular, for a generalized Artin theorem, we give a finite list of
moves to surfaces braids such that equivalent surface braids are related by a finite
application of moves taken from this list, and we give examples of the Alexander
isotopy. We close the chapter with a discussion on a homotopy theory interpretation
of the surface braids.
Chapter 4 contains material that contrasts the knotted surface case with the
classical theory. We show that not all generic surfaces lift to embeddings. Triple
point smoothing and applications thereof are given. We define signs and colors of
triple and branch points, and relate them to the normal Euler number. Cancellation
of cusps and branch points on the projections are discussed. Some of the work in
this chapter is joint work with Vera Carrara.
Chapter 5 contains methods for computing the fundamental group and a pre-
sentation matrix for the Alexander module of the knotted surface. We give explicit
computations for several examples. The chapter closes with a description of the
Seifert algorithm for knotted surfaces. Such an algorithm was used by Giller [Gi] in
the case that the projection had no triple points. We developed the full algorithm
in [CS2] and constructed Heegaard diagrams using our algorithm; Kamada wrote
a version of the algorithm in the surface braid case. We use Kamada's approach
to give a Heegaard diagram of the Seifert solid in the case the surface is given in
Chapter 6 is a review of the algebraic and categorical aspects of knotted sur-
faces. We present solutions to the equations that are generalizations of the Yang-
Baxter equation. Our solutions are based on diagrammatic methods and provide a
good testimony to the power of these methods. The definition of a braided monoidal
2-category with duals (as given in [KV2], [BN] and [BL]) is sketched. We indicate
that embedded surfaces in 3-space form a monoidal 2-category with duals while
surfaces embedded in 4-space form a braided monoidal 2-category with duals. The
chapter closes with the result of Baez and Langford that states that embedded
surfaces form a free braided monoidal category with duals on one self dual object
generator. This result forms the backbone of the future search for invariants that
are analogous to the Jones polynomial.
Some of the exercises are labeled research problems. That means that we do
not know the outcome of the research. If the reader finds a solution before we do,
then that is great!
Throughout this book, the term the classical case refers to the theories of
knotted and linked circles in 3-dimensional space or planar diagrams thereof. All
manifolds and maps are smooth, and 4-space has the standard smooth structure.