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Knot Theory and Its Applications

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When I was about ten years old, my parents used to take my sister and me to visit our Aunt Faygie, and the first thing we did upon arrival was to make a beeline for her jewelry box, placed in a special place just for us. We’d empty its contents onto her living room rug and then pick out our favorite, a very long strand of cheap beads. We’d then go about the business of making all sorts of designs with it. Sometimes the designs got rather complicated, meaning lots of crossing points. But I now realize that, because the strand had no clasp to un-clasp, our designs were always the trivial knot. I’m sure that, had the strand possessed a clasp, we would have wound up with many non-trivial knots, including perhaps K(5; 2) which appears on page 321 of our knot theory book, or the less symmetric knot top-left of p. 228. We also played Cat’s Cradle, another game which resulted in trivial knots. For non-trivial knots, we had to wrap packages or tie our shoes.

Other than those childhood experiences (and tying shoes and ribbons), this is the first time I’ve ever had anything to do with knot theory. (I’ve reviewed books here on things like Galois Theory and Category Theory, but knot theory is new to me.) It was for that reason that I grabbed the opportunity to review this book, and I’m still enthusiastic.

However, I have to be honest and say that there’s much that I don’t understand. I certainly know more about knot theory than I did before (and still more after having read the book twice, and looked up knot theory on Wikipedia), but I have to make the disclaimer that I’m far from an expert. In some sense, of course, that makes me particularly qualified to write this review. After all, it’s written for people like me, who didn’t know knot theory. So I’m coming from the right perspective.

Once I decided to free myself from the expectation (and burden) of thoroughly understanding every single sentence in this book, I enjoyed it immensely. (I’m the kind of math enthusiast who sometimes sits through a colloquium, understanding very little, yet feeling enthralled.)

Besides, there are such pretty pictures! Approximately two-thirds (I counted.) of the pages contain these visions of “complicated simplicity”, and that includes distinct logos at the beginning of each chapter. I knew enough knot theory (and knots in general) to realize that pretty pictures in a knot theory text would be mandatory, so I was not surprised. Still, a lesser book would possibly not sport diagrams of such quantity and quality.

In general, the author strives for clarity, and that was appreciated by this reviewer and will be appreciated by students. The approach he has chosen is the intuitive one. For example, on p. 154, beginnng to describe Dehn surgery: “Suppose that K is an (oriented) knot… Now, “slightly” thicken K, so that we form a 3-dimensional manifold called the tubular neighbourhood of K…” That gives us the idea without bogging us down with the rigors of topology. On the next page he continues with his non-rigorous description; for example, “We may view l as being ‘parallel’ to the knot K”. (He does give the more rigorous definition of l as “the intersection between the Seifert surface of K and T”.)

I also enjoyed how he always keeps us abreast of the general picture, in particular keeping us up to date with respect to the various new results and successes (and failures). For example, again on p. 154: “…every closed connected orientable 3-manifold can be constructed by either of these methods [Dehn surgery and covering spaces]. Therefore, in theory, we should be able to gain insight into the classification of 3-manifolds by using knot theory. However, at present the relationship is weighted in the opposite direction, more has been gleaned about knots by understanding their related 3-manifolds.”

The book also mentions many unsolved, simple-sounding problems. In fact, we don’t have to read far to find them; on p. 24, in a short section on “The cobordism group of knots” (a way to “change” the semi-group of knots into a group), we read that “at the time of writing, no methods have been found that will detect exactly which knots are slice knots” (meaning, which knots are equivalent of the trivial knot — meaning, in turn, the nature of the identity equivalence class).

This book covers a lot of territory. Here’s a rundown of the fifteen chapters: Chapter 1 covers “The Fundamental Concepts of Knot Theory”, including a non-rigorous definition, two descriptions (one more rigorous than the other) of when two knots are equivalent, links (unions of more than one knot), the “sum” of two knots (but no additive inverse), and, again, thoughts on how to “make” an additive inverse (via replacing knots by equivalence classes of knots).

Chapter 2 concerns progress towards classifying knots, via regular diagrams, knot tables, and knot graphs. Here we learn about alternating knots (Non-alternating knots are a relatively new phenomenon.), meaning that, as we move along the knot, the “sense” of each crossing point is the opposite from the preceding. Chapter 3 talks about Global (concerning the set of all knots) versus Local problems (about particular knots).

Chapter 4 chronicles one of the main thrusts of knot theory research, namely finding “knot invariants”. A knot invariant is a function whose domain is the set of knots, and which gives the same value for equivalent knots. (Thus, when we find two knots with different values of the invariant, we know that the knots cannot be equivalent. Knot invariants can distinguish between knots, and which knots can be distinguished depends on the particular invariant.) Examples of knot invariants are minimum number of crossing points, bridge number, unknotting number, linking number, and coloring number.

One of my favorite theorems in the book appears in the first section of Chapter 6, on Seifert Matrices. This theorem concerns something called Seifert surfaces; every knot is the boundary of some surface, called a Seifert surface of the knot. Seifert matrices are a little more involved, but they lead, in the next chapter, to two further invariants, namely the Alexander-Conway polynomial and the signature of a knot. Chapter 7 involves tori, and knots thereon (called, appropriately, torus knots); there turn out to be invariants which apply to torus knots, and not necessarily to all knots.

I was also fascinated by a theorem which I had not known (or not remembered): The complement (in the open 3-ball) of the interior of a torus is a torus; thus the open 3-ball can be formed by the gluing together of two tori — and (just as fascinating) — the “meridian” of one is the “longitude” of the other (and vice versa).

There are at least two ways in which every knot leads to a manifold (and vice versa), and Chapter 8 is about the above-mentioned Dehn surgery and covering spaces. Closely related to knots are tangles (Chapter 9) and braids (Chapter 10, a favorite of mine).

There is another polynomial, and then a class of polynomials, which lead to further invariants; the polynomial which started all this is the Jones polynomial, and so important is it in the history of knot theory that the phenomenon is called “the Jones revolution”.

The scope of this book is such that, not only does it include (as its title implies) applications, but it goes into detail. I confess that I understood very little of the three application chapters, having little background (or memory) of the relevant fields of application. Chapter 12, “Knots via Statistical Mechanics,” talks about “the 6-vertex model” and “the partition function for braids”; it all leads to yet another invariant. Chapter 13, “Knot Theory in Molecular Biology” talks about “DNA and knots,” “site-specific recombination,” “a model for site-specific recombination,” and “recombination due to the recombinase Tn3 Resolvase.” Chapter 14, “Graph Theory Applied to Chemistry” deals with “an invariant of graphs: the chromatic polynomial,”Bing’s conjecture and spatial graphs,” and “the chirality of spatial graphs”.

Finally, Chapter 15 concerns “singular knots”, meaning knots that cross themselves (not merely “over” or “under” but THROUGH). In this chapter, “Vassiliev Invariants” generalizes, in a sense, some of the “polynomial-type” invariants, thus providing a new setting in which to study them as well as giving ”a certain topological interpretation to the Jones-type invariants” (p. 300).

As I said at the beginning of this review, I found this first reading about knot theory tough-going, and there are admittedly parts that I gave up on (mostly the applications). However, I do have some suggestions about some of the things that I did seem to have a good grasp on. (All these suggestions are nitpicks, and exceptions.) On page 35, in explaining how to obtain a graph from any given knot, there could have been an intermediary diagram between Figures 2.3.1(a) and 2.3.1(b); in this diagram could have appeared, in each “white” space, the graph-vertices, and these graph vertices could have then been connected to each “nearly” knot crossing point. It might also be good to say that each c (crossing point) gives rise to an edge of the graph G (and vice versa). (If I have that correct, I think that would be helpful to readers.) And concerning the top diagram on page 77, I’m still confused. Aren’t there two ways to splice, or is he saying always do it vertically ?

Pages 80–81 also stymied me. First he states the theorem that “a closed (i.e., one that is compact and without boundary) orientable surface, F, is topologically equivalent… to the sphere with several handles…” Then, in the next paragraph (after the Example) he says, “Since F has a boundary, then by the above theorem F is homeomorphic to a sphere with several handles…” I need him to explain.

On page149, Example 7.5.1, the calculation seems messed up; the sequence of formulas used seems incorrect to me. (The ideas are quite clear.) Finally, on p. 205, where he explains how to express each braid as a product of the n–1 “special braids”, it would have saved me some groping if he had said, referring to Figure 10.2.9, something like “moving from the bottom of the square to the top”.

But again these are mere occasional nitpicks, and perhaps not all of them well-taken on my part.

This book makes me see that knot theory is almost too pretty and too profound to be true! So much other math comes into it — matrices, determinants, polynomials, continued fractions — even e. It’s an amazing field, connected with other amazing fields, and the book does it all justice.

Marion Deutsche Cohen teaches mathematics at Arcadia University in Glenside, PA. Her poetry book, Crossing the Equal Sign, is about the experience of mathematics.

Date Received: 
Tuesday, October 16, 2007
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Kunio Murasugi
Modern Birkhäuser Classics
Publication Date: 
Marion Cohen

Introduction.- Fundamental Concepts of Knot Theory.- Knot Tables.- Fundamental Problems of Knot Theory.- Classical Knot Invariants.- Seifert Matrices.- Invariants from the Seifert matrix.- Torus Knots.- Creating Manifolds from Knots.- Tangles and 2-Bridge Knots.- The Theory of Braids.- The Jones Revolution.- Knots via Statistical Mechanics.- Knot Theory in Molecular Biology.- Graph Theory Applied to Chemistry.- Vassiliev Invariants.- Appendix.- Notes.- Bibliograph.- Index.

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Monday, June 9, 2008