21 comments on “The revenge of the Perko pair

  1. 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.

  2. 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.)

  3. 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.

  4. 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.

  5. 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.)”

  6. 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”.

  7. Ken Perko (the one who studied law at Harvard; not the one who worked secretly for the Air Force on stealth bombers, drones and the "star wars" project) on said:

    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.

  8. 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).

  9. 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.

  10. I guess “The Math Team” at Wolfram Web is still on the floor with ropes trying to show that 10-161=10-163.

  11. 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.

  12. 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.

  13. 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!

  14. Pingback: Carnival of Mathematics 114 :: squareCircleZ

  15. 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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