The following post recycles Richard Elwes’s lovely blog post and this MathOverflow answer. It is dedicated to the memory of the greatest knot-shaker I have met, Kumar Pallana (1918-2013).

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 [1].

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 [5]. 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 [3], Conway [1], and Rolfsen-Bailey-Roth [6], 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 [4], and incorporated by Dehn and Heegaard in their important survey article on “Analysis situs” in the German Encyclopedia of Mathematics [3].

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!

Perko continues:

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.

Whoa!!

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).

Aspiring topologists disillusioned by the debunking of this myth may still take comfort in the astute observation made elsewhere on the internet: “If a lawyer can do research in knot theory, it can’t be that hard.” Non-cyclic coverings are tangible constructs, understandable with a minimum of group theoretic background. And they don’t just sit there flat on the page like all that boring algebra. But please, QUORA, let’s keep “The Weisstein Pair” out of the hands of children!

Comment by Ken Perko — November 13, 2013 @ 8:21 am |

To see that the Weisstein pair is two inequivalent knots try to give them each a Fox 7-coloring, i.e. an assignment to each segment of an integer from 1 to 7 such that at each crossing the sum of the underpasses equals twice the overpass (mod 7). There are only 8 possibilities. One will work for Weisstein’s so-called 10-162. None work for the Perko knot.

Comment by Ken Perko — November 18, 2013 @ 2:06 am |

Thank you very much for your lovely letter!! I didn’t know most of this- in particular that it was not Reidemeister, but rather Bankwitz and Schumann, who completed the classification of knots with nice crossings.

Comment by dmoskovich — January 24, 2014 @ 4:14 am

Following the same line of reasnoning: Cayley was a lawyer and he invented homological algebra, so homological algebra can’t be that hard.

Comment by Bobito — November 28, 2013 @ 3:10 am |

The reasoning is not that it’s easy because a lawyer can do it. It’s just plain easy, and in this case a high school student can do it. Unlike all that horrible algebra, of which Cayley was certainly a major perpetrator, geometry will often meet you more than half way when you’re trying to solve a difficult problem.

Comment by Ken Perko — December 2, 2013 @ 6:26 am

By the way, Bobito, have you seen the proof of Pythagoras’ theorem that draws a couple of squares with A+B on each side and cuts them up in different ways to demonstrate, at first sight, that A squared plus B squared equals C squared? That’s really all that I do with non-cyclic covering spaces. My wife, who’s an architect, understands this easily and says its not mathematics. She may be right.

Comment by Ken Perko — December 13, 2013 @ 9:48 am |

Thank you for your beautifully written blog! I think the English translators of “Knotentheorie” contributed to the confusion by screwing up Reidemeister’s final footnote. It was the same as the one before it: “1 BANKWITZ, C.: Nicht veroffentlicht” but they chose (without informing the reader) to substitute their own silly judgment of what he meant by referring to a paper already listed in the book’s “Literaturverzeichnis” as number 8 — and published in 1930!

Comment by Ken Perko — January 24, 2014 @ 8:26 am |

Naming them “Perko knots”, Slavik Jablan and Radmilla Sazdanovic have indentified 28 more prime knots with 12 or fewer crossings for which writhe is not invariant. Cf. “LinKnot” (World Scientific, 2007) p.44 and “Diagramatic knot properties and invariants” in “Introductory Lectures on Knot Theory” (id., 2012) p.165. Drawings exhibiting ambient isotopies of these examples, from one writhe to the other, are available on email request to me at “lbrtpl@gmail.com.” Why writhe is invariant for so many other non-alternating knots remains to be explained.

Comment by Ken Perko — August 9, 2014 @ 5:18 pm |

A really interesting Perko knot is illustrated by Mark E. Kidwell & Alexander Stoimenow at page 10 of Michigan Math. J. 51 (2003) — 16 crossings (number 1,184,186), three different writhes, and it’s amphicheiral. Try to show those two drawings are equivalent!

Comment by Ken Perko — August 27, 2014 @ 4:52 am |

Actually, it appears that Bankwitz, not Reidemeister, completed the classification of knots through 9 crossings sometime in or before 1931. Reidemeister gives him full credit for this in the second last sentence of his paper “FUNDAMENTALGRUPPE UND ÜBERLAGERUNG VON MANNIGFALTIGKEITEN”: “Durch die Verkettungszahlen dieser Kurven lässt sich die Klassifikation der Tait-Kirkman’schen Knotentabelle aller Knoten bis zu 9 Ueberkreuzungen, wie Herr BANKWITZ zeigte, bis zu Ende durchführen.”

Comment by Ken Perko — September 29, 2017 @ 11:38 am |

Thanks!

Comment by dmoskovich — October 17, 2017 @ 1:23 pm |

This unambiguous tribute to Bankwitz was watered down somewhat at the end of Reidemeister’s book. The fact that Bankwitz’s paper did not appear until 1934 didn’t help. In reviewing it, Reidemeister observed that it’s methods (linking numbers between branch curves) stemmed from his (Reidemeister’s) own early papers in the 1920s. –Ken Perko

Comment by Ken Perko — February 20, 2018 @ 1:11 am |

Thanks for this update. It’s interesting that some of these ideas go back so far before the “standard” references!

Comment by dmoskovich — February 28, 2018 @ 3:52 pm |

Historians tend to be nearsighted.

Comment by Ken Perko — February 28, 2018 @ 6:29 pm