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.
Knot table up to 7 crossings, from Wikipedia
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[1] 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:
Warning: Not the Perko pair!
This error has infected, most likely among many other places, Wolfram Mathworld and Mathematics 1001 by myself. And guess who pointed this error out…
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.”
[1] Note the corrected date, usually given as 1974.
Simple City 
Great job! I couldn’t have said it better myself.
Thanks! Glad you’re happy with it.
Great entry. It’s nice to hear the correct version of a story that gets passed around a lot. “Amateur” mathematician or not, it was a great catch.
I have a picture of the Perko pair in my book Gauge Fields, Knots and Gravity. Now I’m wondering if it’s correct or not. (It’s not a mere reproduction of either of these.)
Baez’ picture is probably o.k. Only the second drawing in Maths 1001(which is still on display on the Wolfram Web Perko Pair page!) is wrong. There are lots of different (and accurate) drawings in knot theory textbooks by, for example, Adams, Cromwell, Kaufman, Lickorish, Livingston, Manturov, Murasugi and, most recently, Chumtov et al. Also, Eric Weisstein got it right in his 1999 encyclopedia, long before his colleagues at MathWorld decided to play around with a new drawing for 10-162.
Thanks! I believe I got my drawing from a book by Kauffman, either On Knots or Knots and Physics. So, it’s probably okay.
If taken from On Knots (P.U.Press,after all)it’s definitely okay. It’s not in Knots and Physics, which is still more fun to read.
John Baez left this comment on my Google+ page yesterday – I think intending it for here:
“Thanks – I think my book is okay, then… amusingly, in the first edition my coauthor drew the Borromean rings wrong, so that they remained linked when you removed any one. (But I didn’t catch it, so I too am to blame.)”
I suggest that the mismatch of Rolfsen’s 10-161 with 10-163, which after more than two months is still erroneously exhibited on the Wolfram Web sites, should be referred to hereafter as “The Weisstein Pair”.
Here’s another comment on what I like to call “The Weisstein Pair” of knots – 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.
Alexander Stoimenow and Morwen B. Thistlethwaite have each kindly reminded me that the latter proved long ago that the writhe of a knot diagram is a slightly variable invariant fof non-alternating knots, and that for the actual Perko Pair (10-161 and 10-162 in Rolfsen’s table) it must fall (absolutely) between 8 and 10 for 10-crossing diagrams and 7 and 11 for diagrams with 11 crossings. See almost any serious textbook on knot theory discussing discovery of the Jones polynomial and subsequent related work toward the end of the last century.
The phony “10-162” on Wolfram’s mathematically misbegotten “Perko Pair” page (which, as Elwes correctly points out, actually illustrates the mirror image of Rolfsen’s 10-163)has writhe 3, as any non-specialist can see by giving it an orientation and adding up its signed crossings.
Those who pay good money for the mathematical wisdom of a source like that should consider taking their business elsewhere.
I should add, as a lawyer, that nobody has to take my word, or that of Stoimenow and Thistlethwaite, about possible writhes for diagrams of the Perko pair knots; Lickorish spells it out clearly (and authoritatively) at page 47 of “An Introduction to Knot Theory” (Springer-Verlag, 1997).
Disappointed Wolfram Web customers seeking an authentic replacement Perko Pair are welcome to use my own preferred diagrams, illustrated above. Should they choose to shop around for other alternatives, we strongly recommend that they check to be sure that both examples are Khovanov homology thin.
Haha – I’m glad I managed to avoid having the “Elwes pair” hung around my neck… Poor old Eric Weisstein…!
I guess “The Math Team” at Wolfram Web is still on the floor with ropes trying to show that 10-161=10-163.
I see that QUORA shows that “Weisstein Pair” on its “How would you explain knot theory to a 10 year old?” website, suggesting that “they might be delighted to work on the Perko Pair”. This is child abuse! Those who voted that answer up are advised that I am taking names.
QUORA now offers the best of all possible illustrations of authentic Perko Pair knots, explaining knot theory in a way that a 10 year old can understand.
Your blog was forwarded to me by Perko. A nice account explaining a bit of knot theory. But more importantly how not only mathematicians but journalists can screw things up. We mustn’t take ourselves too seriously!
Agreed! And, as in life, the screw-ups are much of the fun!
Agreed here too!
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And speaking of screw-ups, note that the cited “Carnival of Mathematics” Pingback illustrates the wrong knot, declaring it wrong when it’s not.
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Actually, Rolfsen did not renumber his original knot10-163. It and both versions of the Perko pair (10-161 and 10-162) appear separately in all three editions of his wonderful book.