14th August, 2013
The Perko Pair I: the story so far
In the late 19th century, mathematicians and physicists started producing tables of knots. The idea here is that some knots are genuinely different from each other, while others can be deformed to match each other without cutting or gluing, making them essentially the same thing from a topologist’s point of view. The trouble is whether or not two knots are really the same is extremely hard to tell on first sight…. as we shall see.
The tables begin with the unknot, the only knot with no crossings. Then comes the trefoil or overhand knot, which (so long as we don’t distinguish between it and its mirror image) is the only knot with three crossings. There’s similarly one knot with four crossings, then two with five, and so on. In the late 19th century Peter Guthrie Tait and Charles Little got as far as listing the knots with 10 crossings.
However the early knot tabulators (unsurprisingly) made a few errors, and it was not until 1976 that Dale Rolfsen put together a comprehensive list of the knots with up to 10 crossings, based on earlier efforts by John Conway, in turn building on the work of James Alexander in the 1920s. Of the ten-crossing knots they counted 166 separate varieties. (Today’s mathematicians have made it as far as 16 crossings.)
The surprise was that Rolfsen and Conway had also made a mistake! Even armed with 20th century techniques of algebraic topology, a duplication had slipped through. In 1973 Kenneth Perko had been studying a 19th century table of Little, and had realised that two ten-crossing knots (subsequently labelled by Rolfsen as 10161 and 10162) were actually the same thing in disguise.
This story has a very clear moral: telling knots apart is difficult. Really difficult. Even after 75 years of brainpower, mathematicians were still coming badly unstuck, and who knows how long it might have taken without Perko’s alertness. And of course, this just for knots with a paltry ten crossings. Imagine trying to decipher knots with thousands…
Now for the sequel:
The Perko Pair II: Perko strikes again
The above story has been told countless times, and is usually accompanied by a picture of the offending pair of knots, something like this:
The explanation of the mistake is that two updated versions of Rolfsen’s 1976 table are now in circulation. In both, 10162 has been deleted as it should be. But one version (occurring for instance in the latest editions of his book) keeps his original numbering up to 10166 with a space between 10161 and 10163. In the other, see here for example, Rolfsen’s 10163 has been renumbered as 10162, and 10164 as 10163, etc., thus counting up to 10165.
So the knot numbered 10162 in the picture above was actually Rolfsen’s original 10163, and thus not equivalent to 10161! For the avoidance of doubt, here is the real Perko pair, with drawings provided by the man himself. Accept no imitations!
While we’re at it, let’s also put paid to the to the idea that Ken is an ‘amateur’ mathematician. Before going on to a career in law, he says “at Princeton I studied under the world’s top knot theory topologists (Fox, Milnor, Neuwirth, Stallings, Trotter and Tucker)”. I apologise for suggesting otherwise. At least I didn’t pronounce him dead unlike certain other popular maths authors… Also, he adds “That stuff about rope on the living room floor is pure internet nonsense. I did it with diagrams on a yellow legal pad. Ropes wouldn’t work anyway since the two knots are non-isotopic mirror images of each other.”
 Note the corrected date, usually given as 1974.