Yesterday I received correspondence from a certain Kenneth A. Perko Jr., whose name perhaps you have heard before. Its contents are too delicious not to share- knot theory’s favourite urban legend is completely false!
Myth: Ken Perko, a New York lawyer with no formal mathematical training, was having a slow day at the office. Bored and in-between troublesome clients, he toyed with a long piece of rope, which he had tangled up to represent knot in Rolfsen’s table (Rolfsen, like Kuga, was popular among non-mathematicians at the time). As Perko played with it, the knotted rope began to change before his eyes, and glancing back at the book, he suddenly realized that what he was holding in his hands was the ! Was it magic? Ken Perko shook the rope, and did it again. Sure enough, the and were the same knot!
Excited, Ken Perko shot off a paper to PAMS, containing only a title and a list of figures demonstrating an ambient isotopy. His paper entered the Guiness Book of World Records as the “shortest mathematics paper of all time”, and Ken Perko obtained immortality.
This is the Perko pair:
What a story! The human drama, the “math for the masses” aspect that a complete amateur could make a massive mathematical discovery by playing with some string, the beautiful magenta pair of knots, the importance of attention to detail and using all your senses (not just your head)! What a shame that virtually everything written above turns out to be false!
Fact: Ken Perko is a notable knot theorist (without a PhD). At Princeton, he studied, I believe as an undergraduate, “under the world’s top knot theory topologists (Fox, Milnor, Neuwirth, Stallings, Trotter and Tucker)”. As his senior thesis topic, Fox suggested to Perko that he “read the last section of Reidemeister’s book and see if you can figure out how he calculates those linking numbers.” This led to a number of fascinating results, as Perko became a foremost world authority on linking numbers in non-cyclic covering spaces of knot complements. Such invariants, unfashionable as they are, can outperform the Jones polynomial in distinguishing knots, as pointed out by Birman .
By now, Ken Perko was studying law at Harvard Law School (he quotes Reidemeister- “das mathematische Denken its nur der Anfang des Denkens”, or “mathematical thought is just the beginning of thought”). But he was still deeply interested in low dimensional topology, frequently publishing notable results. How cool is that! In 1973, noting that his covering linkage invariants did not distinguish the from the , he pulled out his yellow legal pad and worked out an explicit sequence of Reidemeister moves relating the two diagrams (or rather, one with the mirror image of the other). His celebrated paper on the topic was rejected by TAMS for being too short (poor TAMS), but was picked up by PAMS . It is in fact 2 pages long (plus tables).
What makes this story even more interesting is that the Perko pair in fact falsified what was at the time a commonly-accepted “theorem” of Little, which had been quoted as fact for almost a century. Perko explains:
The duplicate knot in tables compiled by Tait-Little , Conway , and Rolfsen-Bailey-Roth , is not just a bookkeeping error. It is a counterexample to an 1899 “Theorem” of C.N. Little (Yale PhD, 1885), accepted as true by P.G. Tait , and incorporated by Dehn and Heegaard in their important survey article on “Analysis situs” in the German Encyclopedia of Mathematics .
Little’s `Theorem’ was that any two reduced diagrams of the same knot possess the same writhe (number of overcrossings minus number of undercrossings). The Perko pair have different writhes, and so Little’s “Theorem”, if true, would have proven them to be distinct!
Yet still, after 40 years, learned scholars do not speak of Little’s false theorem, describing instead its decapitated remnants as a Tait Conjecture- and indeed, one subsequently proved correct by Kauffman, Murasugi, and Thislethwaite.
I think they are missing a valuable point. History instructs by reminding the reader not merely of past triumphs, but of terrible mistakes as well.
And the final nail in the coffin is that the image above isn’t of the Perko pair!!! It’s the `Weisstein pair’ and mirror , described by Perko as “those magenta colored, almost matching non-twins that add beauty and confusion to the Perko Pair page of Wolfram Web’s Math World website. In a way, it’s an honor to have my name attached to such a well-crafted likeness of a couple of Bhuddist prayer wheels, but it certainly must be treated with the caution that its color suggests by anyone seriously interested in mathematics.”
The reason for this error was the the was deleted from subsequent editions of Rolfsen’s knot tables, so the there is actually . Tamper with knot numberings at your peril!
The real Perko pair (accept no imitations!) is this:
1. J.S Birman, On the Jones polynomial of closed 3-braids, Inventiones Mat. 81 (1985), 287-294 at 293.
2. J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, Proc. Conf. Oxford, 1967, p. 329-358 (Pergamon Press, 1970).
3. M. Dehn and P. Heegaard, Enzyk. der Math. Wiss. III AB 3 (1907), p. 212: “Die algebraische Zahl der Ueberkreuzungen ist fuer die reduzierte Form jedes Knotens bestimmt.”
4. C.N. Little, Non-alternating +/- knots, Trans. Roy. Soc. Edinburgh 39 (1900), page 774 and plate III. This paper describes itself at p. 771 as “Communicated by Prof. Tait.”
5. K.A. Perko, Jr. On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262-266.
6. D. Rolfsen, Knots and links (Publish or Perish, 1976).