x PREFACE

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

braid form.

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.