Perko pair - What's the handedness of these pictures?












4












$begingroup$


In 1974, a paper titled On the Classification of Knots by Ken Perko appeared showing that the knots $10_{161}$ and $10_{162}$ in Dale Rolfsen's knot table were actually the same knot. He included this picture, showing how to deform one into the other:





Edit: A clearer, color-coded version



From that point on, $10_{161}$ and $10_{162}$ became known as the Perko pair, or the Perko knot.



Another view of the pair can be found on the KnotPlot website. That link shows its own explicit deformation between $10_{161}$ and $10_{162}$, and its pair looks like this:





Wikipedia also has pictures of the pair (click here or here for bigger images):





Here's a version that Perko himself drew:





There's a problem, though. I mean, for one thing, these all kinda look nothing like each other. But a bigger problem is this: the Perko knot is chiral! That is, there's a left-handed and right-handed version.



I've drawn my own projection of the Perko knot:





and I'm fairly certain that the one that I drew matches the handedness found in the original paper, as well as the KnotPlot one. However, Wikipedia's first image (the one of $10_{161}$) seems to be a mirror version.



So, my question is this:



Call the one that I drew the left-handed Perko knot, and its mirror image the right-handed Perko knot. What is the handedness of Wikipedia's second image? What are the handednesses of the ones that Perko drew? And am I right in saying that the paper's image and KnotPlot's image are both left-handed, and that Wikipedia's first image is right-handed?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here.
    $endgroup$
    – Akiva Weinberger
    Dec 12 '18 at 16:04






  • 1




    $begingroup$
    The signature of either Perko knot is nonzero. So one could use that to distinguish between the right and left hand versions. Additionally, the Jones polynomial of either Perko knot is not symmetric under $tmapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well.
    $endgroup$
    – Adam Lowrance
    Dec 12 '18 at 16:35










  • $begingroup$
    A Great Lakes sailor has told me that he'd probably describe it as a "double bowline-Ashley's" or maybe just a "double Ashley's".
    $endgroup$
    – Akiva Weinberger
    Dec 16 '18 at 16:36


















4












$begingroup$


In 1974, a paper titled On the Classification of Knots by Ken Perko appeared showing that the knots $10_{161}$ and $10_{162}$ in Dale Rolfsen's knot table were actually the same knot. He included this picture, showing how to deform one into the other:





Edit: A clearer, color-coded version



From that point on, $10_{161}$ and $10_{162}$ became known as the Perko pair, or the Perko knot.



Another view of the pair can be found on the KnotPlot website. That link shows its own explicit deformation between $10_{161}$ and $10_{162}$, and its pair looks like this:





Wikipedia also has pictures of the pair (click here or here for bigger images):





Here's a version that Perko himself drew:





There's a problem, though. I mean, for one thing, these all kinda look nothing like each other. But a bigger problem is this: the Perko knot is chiral! That is, there's a left-handed and right-handed version.



I've drawn my own projection of the Perko knot:





and I'm fairly certain that the one that I drew matches the handedness found in the original paper, as well as the KnotPlot one. However, Wikipedia's first image (the one of $10_{161}$) seems to be a mirror version.



So, my question is this:



Call the one that I drew the left-handed Perko knot, and its mirror image the right-handed Perko knot. What is the handedness of Wikipedia's second image? What are the handednesses of the ones that Perko drew? And am I right in saying that the paper's image and KnotPlot's image are both left-handed, and that Wikipedia's first image is right-handed?










share|cite|improve this question











$endgroup$












  • $begingroup$
    Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here.
    $endgroup$
    – Akiva Weinberger
    Dec 12 '18 at 16:04






  • 1




    $begingroup$
    The signature of either Perko knot is nonzero. So one could use that to distinguish between the right and left hand versions. Additionally, the Jones polynomial of either Perko knot is not symmetric under $tmapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well.
    $endgroup$
    – Adam Lowrance
    Dec 12 '18 at 16:35










  • $begingroup$
    A Great Lakes sailor has told me that he'd probably describe it as a "double bowline-Ashley's" or maybe just a "double Ashley's".
    $endgroup$
    – Akiva Weinberger
    Dec 16 '18 at 16:36
















4












4








4


1



$begingroup$


In 1974, a paper titled On the Classification of Knots by Ken Perko appeared showing that the knots $10_{161}$ and $10_{162}$ in Dale Rolfsen's knot table were actually the same knot. He included this picture, showing how to deform one into the other:





Edit: A clearer, color-coded version



From that point on, $10_{161}$ and $10_{162}$ became known as the Perko pair, or the Perko knot.



Another view of the pair can be found on the KnotPlot website. That link shows its own explicit deformation between $10_{161}$ and $10_{162}$, and its pair looks like this:





Wikipedia also has pictures of the pair (click here or here for bigger images):





Here's a version that Perko himself drew:





There's a problem, though. I mean, for one thing, these all kinda look nothing like each other. But a bigger problem is this: the Perko knot is chiral! That is, there's a left-handed and right-handed version.



I've drawn my own projection of the Perko knot:





and I'm fairly certain that the one that I drew matches the handedness found in the original paper, as well as the KnotPlot one. However, Wikipedia's first image (the one of $10_{161}$) seems to be a mirror version.



So, my question is this:



Call the one that I drew the left-handed Perko knot, and its mirror image the right-handed Perko knot. What is the handedness of Wikipedia's second image? What are the handednesses of the ones that Perko drew? And am I right in saying that the paper's image and KnotPlot's image are both left-handed, and that Wikipedia's first image is right-handed?










share|cite|improve this question











$endgroup$




In 1974, a paper titled On the Classification of Knots by Ken Perko appeared showing that the knots $10_{161}$ and $10_{162}$ in Dale Rolfsen's knot table were actually the same knot. He included this picture, showing how to deform one into the other:





Edit: A clearer, color-coded version



From that point on, $10_{161}$ and $10_{162}$ became known as the Perko pair, or the Perko knot.



Another view of the pair can be found on the KnotPlot website. That link shows its own explicit deformation between $10_{161}$ and $10_{162}$, and its pair looks like this:





Wikipedia also has pictures of the pair (click here or here for bigger images):





Here's a version that Perko himself drew:





There's a problem, though. I mean, for one thing, these all kinda look nothing like each other. But a bigger problem is this: the Perko knot is chiral! That is, there's a left-handed and right-handed version.



I've drawn my own projection of the Perko knot:





and I'm fairly certain that the one that I drew matches the handedness found in the original paper, as well as the KnotPlot one. However, Wikipedia's first image (the one of $10_{161}$) seems to be a mirror version.



So, my question is this:



Call the one that I drew the left-handed Perko knot, and its mirror image the right-handed Perko knot. What is the handedness of Wikipedia's second image? What are the handednesses of the ones that Perko drew? And am I right in saying that the paper's image and KnotPlot's image are both left-handed, and that Wikipedia's first image is right-handed?







knot-theory






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edited Dec 24 '18 at 17:46







Akiva Weinberger

















asked Dec 12 '18 at 16:01









Akiva WeinbergerAkiva Weinberger

14k12268




14k12268












  • $begingroup$
    Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here.
    $endgroup$
    – Akiva Weinberger
    Dec 12 '18 at 16:04






  • 1




    $begingroup$
    The signature of either Perko knot is nonzero. So one could use that to distinguish between the right and left hand versions. Additionally, the Jones polynomial of either Perko knot is not symmetric under $tmapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well.
    $endgroup$
    – Adam Lowrance
    Dec 12 '18 at 16:35










  • $begingroup$
    A Great Lakes sailor has told me that he'd probably describe it as a "double bowline-Ashley's" or maybe just a "double Ashley's".
    $endgroup$
    – Akiva Weinberger
    Dec 16 '18 at 16:36




















  • $begingroup$
    Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here.
    $endgroup$
    – Akiva Weinberger
    Dec 12 '18 at 16:04






  • 1




    $begingroup$
    The signature of either Perko knot is nonzero. So one could use that to distinguish between the right and left hand versions. Additionally, the Jones polynomial of either Perko knot is not symmetric under $tmapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well.
    $endgroup$
    – Adam Lowrance
    Dec 12 '18 at 16:35










  • $begingroup$
    A Great Lakes sailor has told me that he'd probably describe it as a "double bowline-Ashley's" or maybe just a "double Ashley's".
    $endgroup$
    – Akiva Weinberger
    Dec 16 '18 at 16:36


















$begingroup$
Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here.
$endgroup$
– Akiva Weinberger
Dec 12 '18 at 16:04




$begingroup$
Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here.
$endgroup$
– Akiva Weinberger
Dec 12 '18 at 16:04




1




1




$begingroup$
The signature of either Perko knot is nonzero. So one could use that to distinguish between the right and left hand versions. Additionally, the Jones polynomial of either Perko knot is not symmetric under $tmapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well.
$endgroup$
– Adam Lowrance
Dec 12 '18 at 16:35




$begingroup$
The signature of either Perko knot is nonzero. So one could use that to distinguish between the right and left hand versions. Additionally, the Jones polynomial of either Perko knot is not symmetric under $tmapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well.
$endgroup$
– Adam Lowrance
Dec 12 '18 at 16:35












$begingroup$
A Great Lakes sailor has told me that he'd probably describe it as a "double bowline-Ashley's" or maybe just a "double Ashley's".
$endgroup$
– Akiva Weinberger
Dec 16 '18 at 16:36






$begingroup$
A Great Lakes sailor has told me that he'd probably describe it as a "double bowline-Ashley's" or maybe just a "double Ashley's".
$endgroup$
– Akiva Weinberger
Dec 16 '18 at 16:36












3 Answers
3






active

oldest

votes


















2












$begingroup$

The diagrams in On the Classification of Knots are left-handed.



The pictures from KnotPlot are left-handed.



Wikipedia's pictures are right-handed.



The first of Perko's drawings is right-handed; the second is left-handed.



I determined this experimentally by tying the knots in my headphones and manipulating them until I got them into a form I recognized. I didn't compute the Jones polynomials, since that would probably take around an hour per knot.



This was much harder than I thought it would be. I thought that a physical knot would naturally end up in the configuration of "lowest energy" (whatever that means) when the ends are pulled, but it turns out that knots are much more annoying than that.



I have a lot of respect for the early knot theorists who made knot tables before anyone knew how to rigorously prove that any two knots were distinct. And I don't blame them for thinking the Perko pair was two distinct knots.



Here's the (left-handed) Perko knot tied in my headphones: https://imgur.com/a/OC3TAkH






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    The sign of the writhe of any 10-crossing diagram (the sum of the signs of the crossings, which must be either + or - 10 or 8 for the Perko pair knot) will show the handedness of any particular drawing. It's no big deal.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
      $endgroup$
      – Akiva Weinberger
      Dec 24 '18 at 12:12










    • $begingroup$
      I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
      $endgroup$
      – Akiva Weinberger
      Dec 24 '18 at 18:49












    • $begingroup$
      Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
      $endgroup$
      – user43408
      Dec 25 '18 at 2:37





















    0












    $begingroup$

    I like this new diagram of the Perko pair knot; it's in "twisted ribbon" form, which I have not seen before. --Ken Perko






    share|cite|improve this answer









    $endgroup$














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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      The diagrams in On the Classification of Knots are left-handed.



      The pictures from KnotPlot are left-handed.



      Wikipedia's pictures are right-handed.



      The first of Perko's drawings is right-handed; the second is left-handed.



      I determined this experimentally by tying the knots in my headphones and manipulating them until I got them into a form I recognized. I didn't compute the Jones polynomials, since that would probably take around an hour per knot.



      This was much harder than I thought it would be. I thought that a physical knot would naturally end up in the configuration of "lowest energy" (whatever that means) when the ends are pulled, but it turns out that knots are much more annoying than that.



      I have a lot of respect for the early knot theorists who made knot tables before anyone knew how to rigorously prove that any two knots were distinct. And I don't blame them for thinking the Perko pair was two distinct knots.



      Here's the (left-handed) Perko knot tied in my headphones: https://imgur.com/a/OC3TAkH






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        The diagrams in On the Classification of Knots are left-handed.



        The pictures from KnotPlot are left-handed.



        Wikipedia's pictures are right-handed.



        The first of Perko's drawings is right-handed; the second is left-handed.



        I determined this experimentally by tying the knots in my headphones and manipulating them until I got them into a form I recognized. I didn't compute the Jones polynomials, since that would probably take around an hour per knot.



        This was much harder than I thought it would be. I thought that a physical knot would naturally end up in the configuration of "lowest energy" (whatever that means) when the ends are pulled, but it turns out that knots are much more annoying than that.



        I have a lot of respect for the early knot theorists who made knot tables before anyone knew how to rigorously prove that any two knots were distinct. And I don't blame them for thinking the Perko pair was two distinct knots.



        Here's the (left-handed) Perko knot tied in my headphones: https://imgur.com/a/OC3TAkH






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          The diagrams in On the Classification of Knots are left-handed.



          The pictures from KnotPlot are left-handed.



          Wikipedia's pictures are right-handed.



          The first of Perko's drawings is right-handed; the second is left-handed.



          I determined this experimentally by tying the knots in my headphones and manipulating them until I got them into a form I recognized. I didn't compute the Jones polynomials, since that would probably take around an hour per knot.



          This was much harder than I thought it would be. I thought that a physical knot would naturally end up in the configuration of "lowest energy" (whatever that means) when the ends are pulled, but it turns out that knots are much more annoying than that.



          I have a lot of respect for the early knot theorists who made knot tables before anyone knew how to rigorously prove that any two knots were distinct. And I don't blame them for thinking the Perko pair was two distinct knots.



          Here's the (left-handed) Perko knot tied in my headphones: https://imgur.com/a/OC3TAkH






          share|cite|improve this answer









          $endgroup$



          The diagrams in On the Classification of Knots are left-handed.



          The pictures from KnotPlot are left-handed.



          Wikipedia's pictures are right-handed.



          The first of Perko's drawings is right-handed; the second is left-handed.



          I determined this experimentally by tying the knots in my headphones and manipulating them until I got them into a form I recognized. I didn't compute the Jones polynomials, since that would probably take around an hour per knot.



          This was much harder than I thought it would be. I thought that a physical knot would naturally end up in the configuration of "lowest energy" (whatever that means) when the ends are pulled, but it turns out that knots are much more annoying than that.



          I have a lot of respect for the early knot theorists who made knot tables before anyone knew how to rigorously prove that any two knots were distinct. And I don't blame them for thinking the Perko pair was two distinct knots.



          Here's the (left-handed) Perko knot tied in my headphones: https://imgur.com/a/OC3TAkH







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 16 '18 at 15:14









          Akiva WeinbergerAkiva Weinberger

          14k12268




          14k12268























              2












              $begingroup$

              The sign of the writhe of any 10-crossing diagram (the sum of the signs of the crossings, which must be either + or - 10 or 8 for the Perko pair knot) will show the handedness of any particular drawing. It's no big deal.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 12:12










              • $begingroup$
                I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 18:49












              • $begingroup$
                Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
                $endgroup$
                – user43408
                Dec 25 '18 at 2:37


















              2












              $begingroup$

              The sign of the writhe of any 10-crossing diagram (the sum of the signs of the crossings, which must be either + or - 10 or 8 for the Perko pair knot) will show the handedness of any particular drawing. It's no big deal.






              share|cite|improve this answer









              $endgroup$













              • $begingroup$
                Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 12:12










              • $begingroup$
                I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 18:49












              • $begingroup$
                Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
                $endgroup$
                – user43408
                Dec 25 '18 at 2:37
















              2












              2








              2





              $begingroup$

              The sign of the writhe of any 10-crossing diagram (the sum of the signs of the crossings, which must be either + or - 10 or 8 for the Perko pair knot) will show the handedness of any particular drawing. It's no big deal.






              share|cite|improve this answer









              $endgroup$



              The sign of the writhe of any 10-crossing diagram (the sum of the signs of the crossings, which must be either + or - 10 or 8 for the Perko pair knot) will show the handedness of any particular drawing. It's no big deal.







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Dec 24 '18 at 11:46









              user43408user43408

              211




              211












              • $begingroup$
                Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 12:12










              • $begingroup$
                I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 18:49












              • $begingroup$
                Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
                $endgroup$
                – user43408
                Dec 25 '18 at 2:37




















              • $begingroup$
                Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 12:12










              • $begingroup$
                I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
                $endgroup$
                – Akiva Weinberger
                Dec 24 '18 at 18:49












              • $begingroup$
                Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
                $endgroup$
                – user43408
                Dec 25 '18 at 2:37


















              $begingroup$
              Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
              $endgroup$
              – Akiva Weinberger
              Dec 24 '18 at 12:12




              $begingroup$
              Ah, I hadn't thought of checking the writhe… though, to be fair, it doesn't prove which handedness it must be unless we already knew the possible writhes of a minimal diagram of a given handedness of the knot. In any case: Thanks for taking the time to look at my question, though, Perko!
              $endgroup$
              – Akiva Weinberger
              Dec 24 '18 at 12:12












              $begingroup$
              I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
              $endgroup$
              – Akiva Weinberger
              Dec 24 '18 at 18:49






              $begingroup$
              I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/…
              $endgroup$
              – Akiva Weinberger
              Dec 24 '18 at 18:49














              $begingroup$
              Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
              $endgroup$
              – user43408
              Dec 25 '18 at 2:37






              $begingroup$
              Actually, the numerical value of the minimal crossing writhe doesn't really matter from the standpoint of handedness; just whether it's + or -. ("in case"? Shame on those who don't!)
              $endgroup$
              – user43408
              Dec 25 '18 at 2:37













              0












              $begingroup$

              I like this new diagram of the Perko pair knot; it's in "twisted ribbon" form, which I have not seen before. --Ken Perko






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                I like this new diagram of the Perko pair knot; it's in "twisted ribbon" form, which I have not seen before. --Ken Perko






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                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  I like this new diagram of the Perko pair knot; it's in "twisted ribbon" form, which I have not seen before. --Ken Perko






                  share|cite|improve this answer









                  $endgroup$



                  I like this new diagram of the Perko pair knot; it's in "twisted ribbon" form, which I have not seen before. --Ken Perko







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 2 at 5:04









                  user43408user43408

                  211




                  211






























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