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

Question

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 (redrawn by me):

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?

Incidentally, MathWorld's images are incorrect! More details can be found here and here, and here. — Akiva Weinberger, Dec 12 '18 at 16:04
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 $t\mapsto t^{-1}$, and so it can be used to distinguish the right and left hand versions as well. — Adam Lowrance, Dec 12 '18 at 16:35
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". — Akiva Weinberger, Dec 16 '18 at 16:36

score 2 · Accepted Answer · answered Dec 16 '18 at 15:14

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://i.stack.imgur.com/uMFvE.jpg

score 2 · Answer 2 · answered Dec 24 '18 at 11:46

2

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.

answered Dec 24 '18 at 11:46

user43408

12
4

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! – Akiva Weinberger Dec 24 '18 at 12:12
I'll put a link to your Quora response, in case anyone who sees this in the future finds it interesting: https://www.quora.com/How-often-are-new-knots-invented/answer/Ken-Perko/comment/81352245 – Akiva Weinberger Dec 24 '18 at 18:49
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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!) – user43408 Dec 25 '18 at 02:37

score 2 · Answer 3 · answered Jan 02 '19 at 05:04

2

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

answered Jan 02 '19 at 05:04

user43408

12
4

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

3 Answers3