The Hankering (Knots)

July 9, 2010

It was only a matter of time, I suppose, before I felt the need, the yen, the hankering for some mathematical activity again. To that end, I borrowed my girlfriend’s copy of The Knot Book by Colin C. Adams.

It’s about knots.

More specifically, it’s about knot theory, which was a pretty hip subfield of topology last I checked. In other words, studying knots is a serious mathematical pursuit if you do it seriously enough.

I like Adams’ perspective on math as an activity. Here’s how the book starts:

“Mathematics is an incredibly exciting and creative field of endeavor. Yet most people never see it that way. Nonmathematicians too often assume that we mathematicians sit around talking about what Newton did three hundred years ago or calculating a couple of extra million digits of pi. They do not realize that more new mathematics is being created now than at any other time in the history of humankind.”

He basically has me at hello. And I like his idea to use knots as a quick way in. We’re all used to them and have some intuition about them, but even the problem of telling knots apart is still open.

His exposition on knots themselves is promising thus far (I’ve just begun the book). And he includes some great history of where the study came from. In the 1880s, Lord Kelvin hypothesized that atoms might be knotted ether, and his colleague Peter Guthrie Tait started tabulating all possible knots, in the hopes that this would help him understand chemistry in some fundamental way. Of course, the whole notion of ether was disproved before the end of that decade, and scientists left knots behind. Mathematicians picked up the thread, though, and knot theory became a serious field. In the 1980s, it turned out that there are applications in studying knotted DNA molecules. So knots are important in the real world after all.

Which must be nice for Tait, who I imagined felt pretty crummy about having spent years of work fleshing out a totally defunct theory. I wish he could have lived to see the usefulness of his work.

Here’s a fun tidbit from the math: you can add knots together, and even more, you can break knots down into sums of the simplest kinds of knots. Taking our cue from how we do the same when we factor numbers, we call these simplest knots (which can’t be broken down further) prime knots. For your consideration, here are some of the first prime knots. See if you can guess what the numbers below each knot indicate.

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