Why Knot? An Introduction to the Mathematical Theory of Knots

The first book in a new "Math is Fun" series from Key Curriculum Press is a knot theory comic book by Colin Adams, complete with a trefoil-adorned superheroine Knot-man, her boy sidekick Knot-girl, and even a trusty dog Knot-cat. Knot projections are shown on a screen titled "Lord of the Strings" and the Hunchback of Knotre Dame makes a brief appearance. As you can tell, older readers will have to have a high tolerance for knot puns. From my experience, this will not bother students at all. In fact, this is a book which you should purchase for your mathematics seminar room and leave lying around where students will pick it up casually, find it too much fun to stop reading, and discover that knot theory is interesting stuff. I also hope that many copies will somehow make their way into high schools.

Beneath the comic illustrations and the puns and the general fun, Why Knot? is a brief introduction to knot theory, covering roughly the first 40 pages of Adams' classic The Knot Book (with one notable omission I will discuss below) together with a little material on knotted DNA and synthesizing knotted molecules. Students will see the composition of knots, Reidemeister moves, the linking number for links, the crossing number and unknotting number for knots, and an introduction to the Dowker notation for knots.

Knot theory is a hands-on subject which shouldn't just be read about: one would miss too much of the fun. Hence Why Knot? includes 39 experiments for students to try, many of which ask them to manipulate one form of a knot into another, or say which knots are the same and which are different. To help in doing these, the comic book comes packaged with a string of 50 "tangle particles," a tool developed by Zen sculptor Richard Zawitz. They are rather like pieces of plastic macaroni which you can plug into one another to make long skeins which you can tie into knots. They are certainly entertaining, but I find them awkward to work with: the knots you get are orange and purple, wavy, and not terribly easy to manipulate. I tried the experiments with the tangle, but I confess that I usually retreated to a trusty four-foot piece of clothesline tied into a loop. Alas, the inclusion of the tangle raises the price to where you will probably want to buy only a few copies, rather than many, for the seminar room.

In a quick introduction to knot theory, it is not surprising that there are many places where readers are asked to take statements on faith, with phrases like "We don't have time to prove this here, but it seems intuitively reasonable." Indeed there is only one proof in the book: the argument that the linking number is unchanged by Reidemeister moves and hence is a link invariant. However, there is one omission which I think is unfortunate. After the trefoil knot is introduced, it is carefully noted that no amount of fiddling with the knot and not being able to untie it can show that it is not equivalent to the unknot. "So we want a proof that these are distinct knots. This is where the mathematics comes in." Later, we are reminded several times that we still don't really know if there are knots which are not equivalent to the unknot. However, the mathematics never comes in: no proof is ever given.

This is a shame, since there is an easy proof that the trefoil is knotted, which Richard Crowell wrote about for high school students, which is given on pages 22-25 of The Knot Book and which Colin Adams' friend Mel Slugbate has presented in many student talks. It uses the lovely notion of tricolorability: one colors the arcs of a projection of the trefoil with three colors in such a way that at 1) at least two colors are used, and 2) at every crossing, the three arcs meeting at that crossing are either three different colors or all the same color. The existence of such a coloring is unchanged by Reidemeister moves and hence is a knot invariant. Since the unknot clearly does not have such a coloring, the trefoil and the unknot are indeed distinct. I recommend knowing this proof so that you can quickly satisfy frustrated students stopping by from the seminar room, perhaps complaining, "If we don't prove there are any knotted knots, the whole subject might be trivial!"

---

Philip Straffin (straffin@beloit.edu) is Professor of Mathematics at Beloit College. As a pre-teenager he assembled a collection of Superman and Little Lulu comics which would make his fortune at today's prices. As a graduate student he studied knot theory with Ray Lickorish at Cambridge University before straying into algebraic topology at Berkeley and then more distant areas of mathematics at Beloit. He regularly teaches an undergraduate topology course which includes knot theory and a concrete geometric treatment of surfaces and three-manifolds.