The guy in the office next door to me is a knot theorist, or a low-dimensional topologist. Knot theory is rampant, in fact, in an open and path-connected neighborhood of my office, and some of it spills over into my own compact sub-neighborhood from time to time. I must say that is a pretty pleasant experience, be it a matter of eavesdropping or kibitzing. Indeed, last term my knot-theoretical neighbor was running a seminar down the hall in which a certain combinatorial knot invariant needed to be computed, the whole business having been reduced to a question in elementary number theory. As I was ambling by I was corralled by his class and given the problem, which, deo gratias, I could in fact solve and, as Jeeves would put it to Bertie Wooster, I was happy to have given satisfaction. A most enjoyable experience.
And that’s really what the book under review is all about. It is, quite simply, enjoyable to be able to play both ends against the middle when these ends are knot theory and number theory: is there any better example of the ends justifying the means (if I may be forgiven a truly egregious pun)? Both subjects are exquisitely pretty, accessible and yet non-trivial already after turning the first half-dozen pages (or fewer!) of any decent book introducing either. Their interplay is bound to be extremely evocative.
Morishita emphatically takes this position and pleads his case in no uncertain terms from the very outset: his first chapter’s first section is titled “Two ways that branched out from C. F. Gauss — quadratic residues and linking numbers.” I should recommend that the word “way” be interpreted in accord with the Japanese “dō,” which indicates something like a path to be followed toward enlightenment — so it is, for instance, that “judo” translates to “the gentle way” while “jujutsu” translates to “the gentle art.” Morishita presents Gauss as (well, what else?) a visionary par excellence, whose unsurpassed intuition speaks to us still: “Although there seems to be no connection between the Legendre symbol and the linking number at first glance … there is indeed a close analogy …[:] they are defined in an exactly analogous manner …” (cf. p.3, loc.cit.). Morishita then adds: “Since Gauss took an interest in knots in his youth we may imagine that he already had a sense of [this] analogy …”
This is deep stuff, isn’t it? And it’s very, very pretty. On p.58 Morishita in fact presents the dénouement of the foregoing in the form of the following characterization: if q*=(–1)(q–1)/2q, and we write lk(q,p) for a certain well-motivated mod 2 linking number, then the Legendre symbol (q*/p) satisfies the relation (q*/p)=(–1)lk(q,p). Wow!
All right, then, what does it take to get to a result like this? The answer is found in Morishita’s first three chapters, where he addresses the interplay between fundamental groups and Galois groups (shades of Grothendieck, of course), and 3-manifolds and “number rings.”
Under the latter heading we encounter rings Ok of integers in number fields (or, as per Borevich-Shafarevich for example, maximal orders) and a stout discussion of class-field theory, as expected. But we also see connections to étale cohomology, Artin-Verdier duality, and the Tate-Poitou exact sequence. The rationale for this is found in the immediately subsequent discussion of direct analogies between algebraic-topological objects attached to knots and certain mainstays of algebraic and arithmetic geometry. For instance, the fundamental group of the circle (isomorphic to Z) is tied to the fundamental group of Spec(Fq), and so the circle itself is tied to Spec(Fq). Then a knot, as an embedding of the circle in a manifold, is taken as analogous to the embedding of Spec(Fp) in Spec(Ok) for suitable Ok, with k a number field.
A specific instance of this is the parallel between a knot in R3 and the embedding of Spec(Fp) in Spec(Z) with ∞ thrown in, where p is a prime, of course. Finally, a link group, being by definition the fundamental group of a knot complement in a manifold, is tied to a Galois group “with restricted ramification” of the form π1(Spec(Ok)\S), where k is unramified outside S together with the infinite primes of k. Thus, these analogies are not only gorgeous, they’re obviously very deep, and this should be enough of a selling point for this book.
There’s more. Morishita proceeds to talk about the theme of decomposition of primes as well as knots, and thus hits one of the raisons d’être of reciprocity laws, and then turns up the heat even more: Milnor invariants are tied to multiple residue symbols, the ideal class group is given a three-part work-out vis à vis homology groups, all leading to, e.g., a discussion of the Alexander polynomial. We encounter the Iwasawa Main Conjecture, too, and the book’s last two chapters deal with the suggestive themes of “moduli spaces of representations of knot and prime groups” and (watch this!) “deformations of hyperbolic structures and p-adic ordinary modular forms” or, more specifically, of p-adic Galois representations.
Well, it’s clear that this book, a monograph aimed at giving an exposé of a rather young subject, requires a bit of an eclectic background in its readership, and isn’t really meant for the rookie — it’s not a textbook (no exercises, for example),. But once you’ve lived long enough in mathematics, the themes addressed in Knots and Primes: An Introduction to Arithmetic Topology are both familiar and exceedingly attractive. This is a fascinating topic — and then some — and Morishita’s book is an important contribution. Hopefully it will spur a lot of work in this beautiful and accessible area of contemporary mathematics.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.