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Surface-Knots in 4-Space

Seiichi Kamada
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Michael Berg
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Knot theory is very hot stuff these days. When I was a kid, it was a quiet backwater, as the phrase goes, but now it’s a torrent. Perhaps it was the work of Edward Witten in the late 80s, in response to visionary promptings of Sir Michael Atiyah, that began to stir the waters: after all what could be cooler than to tie knot theory to quantum field theory? Maybe it’s the appearance a little earlier of the Jones polynomial (the focus of what Atiyah and Witten were up to): Vaughan F. R. Jones made his wonderful discovery in the context of representations of C*-algebras (more specifically von Neumann algebras) — deep waters! And then there’s the HOMFLY polynomial, whose birthdate also goes back to the middle 1980s — a fabulously popular player: three of my department colleagues are playing this game. Well, from Jones to HOMFLY to TQFT, doubtless it’s the cumulative effect of all these mathematical developments that is responsible for the explosion of activity in knot theory, but indeed (unlike myself) it all dates back only some thirty years or so.

There is of course the added feature that knots and links (deformed circles in 3-space or assemblies thereof) are easy to picture, are apparently easy to draw (although one is readily disabused of this conceit after trying to get the better of a few of the nastier pictures), and are therefore as seductive as, say, elementary number theory. Which youth isn’t struck with amazement at the law of quadratic reciprocity, for instance? But, just as is the case with number theory, it ain’t necessarily so, as the Gershwins put it so pithily. Things get very sticky fast (Klassenkörper theory for example), and doing knot theory right is vouchsafed only to faithful and committed scholars: like doing Zahlentheorie, doing Knoten und Zöpfetheorie is a marathon and takes a lot of conditioning.

Having just juxtaposed number theory and knot theory, let’s talk about Emil Artin, a master of algebraic number theory as well as a pioneer in modern algebraic topology: his book with Hel Braun, Introduction to Algebraic Topology, is still well-worth reading. But I bring him up specifically because he is also a pioneer in knot theory; in fact, even in the subject of surface knots. It is well-known that he was responsible for the presentation of the braid group, which has become such a big part of the entire knotty enterprise. In the Preface to the present book, Katada points out that “the beginning of the study of surface-knots is due to E. Artin in the 1920s,” and this underscores the visionary quality of so much of Artin’s work. (Another example along these lines is his cohomological approach to class-field theory as recorded in the book (or lecture notes) by that title, co-authored with his student, John Tate).

Kamada goes on to note that the next players in the field of research on surface knots were R. H. Fox and John Milnor; more about that presently. In any event, today we are dealing with knots, links, braids, and algebraic objects attached to these, all in the service of low-dimensional topology. It is standard, after all, to consider knots in terms of their set-theoretic complement in the ambient 3-space, or even its 1-point compactification, the 3-sphere. Now with surface knots we are ratcheting the whole business up by one dimension. The plot thickens…

The book under review has all these things: knots, links, and braids are there from beginning to end, but these are not your garden-variety knots: we’re not knotting strings (or 1-manifolds) anymore, now we’re knotting 2-manifolds, i.e. surfaces. We need to do so in 4 dimensions: just think of what a Klein bottle needs in order to survive. And I guess this raises the question of whether it’s necessary to learn, first, about knots in 3-space. Well, I don’t know, but it’s bound to be an excellent idea, if only because God made us three-dimensional, as least spatially, and on general mathematical grounds it is proper to do geometry in progressively higher dimensions. We do plane geometry before we do spatial geometry, for example. But I’ll leave that point aside, except to recommend W. B. Raymond Lickorish’s book, An Introduction to Knot Theory (and, yes, he’s the L in HOMFLY). Kamada’s book is very thorough, however, and his second chapter is expressly devoted to knots (in 3-space, as distinct from 2-dimensional “surface knots” in 4-space), and perhaps that’s enough. But, still, I can’t really imagine going at surface knots without first getting to know the knots that can cause such trouble in my shoe-laces (properly braids, I guess).

As noted above, Kamada continues in his Preface to describe early developments in surface knot theory, stating that Fox introduced the so-called motion picture method and Milnor dealt with knot concordance. A bit later on he says the he, himself, “has been studying surface-knots using 2-dimensional braids since the 1990s.” Furthermore,

[t]heorems on braiding of surface-knots … have been established [and s]ince the late 1990s, invariants pf knots and surface-knots using quandles and their (co-)homology theory have been studied … [T]heir invariants are now extended and generalized to various invariants so that they are used to study chirality of knots, hyperbolic volumes and Chern and Simons’ invariants, invertibility of surface-knots, triple point numbers of surface-knots, etc.

We are dealing with a young(ish) but well-developed subject, with a lot of activity going one. The book under review goes at this material with guns blazing. To wit,

the motion picture method and a method describing surface-knots by classical diagrams with markers and introduced [in Chapter 3, as well as h]ow to compute the knot group of a surface knot from a motion picture … Knot concordance and knot cobordism are discussed in Chap. 7. Chapter 8 is devoted to the study of quandles and colorings of knots and surface-knots … In Chap. 9 we introduce the (co-)homology groups of quandles … [And p]resentation of knots using braids and presentation of surface-knots using 2-dimensional braids are introduced in Chap. 10.

Thus, Kamada provides a discussion of a great deal of important machinery and current approaches to both knot theory in the more familiar and prosaic sense as well as the more exotic surface-knot theory, the book’s main focus. This juxtaposition is certainly the right pedagogical thing to do.

The book is well-written, theorems are plentiful and proven, there are a huge number of diagrams and pictures (how could there not be?), there are lots of examples, and there are even exercises. This book indeed looks like a good place to learn about surface knots in 4-space.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

See the table of contents in the publisher's webpage.