Braid Foliations in Low-Dimensional Topology

It strikes me that this book is a real mitzvah for budding knot theorists and low-dimensional topologists. The claim is that “only mild prerequisites are required, such as an introductory knowledge of knot theory and differential geometry,” and there are over 200 figures. So it’s what you see is what you get, if you’ll pardon the facile play on words. It’s clear however, that, to be sure, some knot theory had better be part of the reader’s arsenal from the outset, but, happily, there are many ways available for a rookie to get what’s needed. The obvious candidates include such texts as Lickorish’s Introduction to Knot Theory and Rolfsen’s Knots and Links; and then there are aleph-null on-line surveys, workshop notes, lecture notes, course notes, and so on: it’s hard to go wrong here. And differential geometry is among those things everyone should know something about and accordingly graduate programs abound with standard courses therein. Or, for self-study, how about the gem by Bott and Tu, Differential Forms in Algebraic Topology, preceded by Tu’s Introduction to Manifolds?

All right then, once this bar has been cleared, what’s a braid foliation? Well, on page 22 of the book under review we find out that it comes about by intersecting a surface, embedded in the 3-sphere, \(S^3\) (low-dimensional topologists like to do this — it’s either that or just 3-space) bounded by a braid with the braid fibration on \(S^3\). Regarding this braid fibration, we learn on pp. 3–4 that the complement of the unknot (i.e. just a circle, modulo isotopy) “fibers over \(S^1\) with \(D^2\) fibres,” whence one naturally gets a family of discs parameterized by (and “on”) \(S^1\), and it is this that constitutes the desired braid fibration of \(S^3\). The upshot is that the result of taking the aforementioned intersection is the braid foliation on the given surface. This is all much more transparent, of course, when one has pictures to mess around with; happily the authors provide them abundantly (as already mentioned above).

Next question: why? Well, the authors say (p. 2) that their “focus … will not be to distinguish between links in distinct isotopy classes [one of the biggest deals in the entire business: it’s not even known whether the redoubtable Jones polynomial tags the unknot!], but rather to investigate the internal structure within a fixed link type.” This is clearly a very good idea and is the right way to start: it’s an excellent way for a relative beginner to get appropriately dirty hands, and to learn some serious knot, link, and braid theory. The authors note that “the interested reader who is familiar with … link invariants is welcome to discover new connections between braid foliations and link invariants.” And there’s the rub, or at least one of them.

Another rub is more intrinsic: the Preface contains the passage(s),

The original idea is due to … Bennequin … in the 1980s … In the 1990s Joan Birman and William Menasco [co-author of the present book] developed and systematized the theory of braid foliations … and used these techniques … to probe the landscape of closed braids representing topological link types with their work culminating in the Markov Theorem without Stabilization…

And there’s more to follow: this is certainly and active area of research.

So, the AMS once more presents the mathematical community with a strong text geared to getting graduate students and other relative beginners into the game. The present book is thorough and well-structured, leads the reader pretty deeply into the indicated parts of knot- and link-theory and low-dimensional topology and does so effectively and (as far as I can tell) rather painlessly. There are exercises at the close of each of the dozen chapters — clearly they should be taken very seriously if a non-trivial attempt at learning this material is to be made. All in all the book looks like a hit.

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