Talk:Torus knot

Don't make people guess what you mean (by p-fold dunce cap)
In the section titled Properties this sentence occurs:

"Let Y be the p-fold dunce cap with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle."

Many people know what a dunce cap is; still it would be nice if it could have been defined in this article.

Almost no one knows what a "p-fold" or "q-fold" dunce cap is. Some people might be able to guess what is meant.

Do not make people guess what you mean. Just say it in unambiguous language!

Especially when it takes almost no more words to say what you mean than to not say what you mean.50.205.142.35 (talk) 00:18, 22 December 2019 (UTC)

Parameter
Probably the parameter range in the parametrization this article begins with should be extended to include &phi; = 0 -- for this just replace the first &lt; with a &le;. 99.60.16.154 (talk) 06:10, 5 July 2010 (UTC)


 * It might be that way so that the tex doesn't blow up, as it does with all but the most basic tex commands (annoying). I just got rid of the \frac's in the parametrization which among other things fixed that. --Vaughan Pratt (talk) 23:41, 13 September 2010 (UTC)

Mismatch between figures and captions
I think the illustrations for the (2,3) and (3,8) case show the mirror image of what they should be. They seem to show the (-2,3) and (-3,8)-torus knots instead. Can someone confirm which is the correct convention? It would be easy to fix either the parametrization of the images. Selinger (talk) 18:43, 13 September 2010 (UTC)
 * How does negating p make any difference? It just reflects the figure about the x-axis but it's already symmetric about it. (But I agree that neither of those figures are torus knots.) --Vaughan Pratt (talk) 23:37, 13 September 2010 (UTC)
 * Vaughan: Last I checked, the left-handed trefoil is not equivalent to the right-handed trefoil up to continuous deformation. The parametrization on this page defines the (2,3)-torus knot as a left-handed trefoil, but the image showed a right-handed trefoil, which is therefore not the (2,3)-torus knot but the (-2,3)-torus knot. So I have updated the image now. I didn't change the image for the (3,8), because it seems correct after all (not sure what I was looking at). Selinger (talk) 01:48, 14 September 2010 (UTC)
 * Oh, that. Sorry, didn't think to look at the parity, I read too quickly and thought you were complaining that it wasn't a metrically accurate picture; in that respect it's different from the one above, which is accurate.  Yes, up to continuous deformation of the complementary space the old figure was for (-2,3), or equivalently (2,-3), so this is a non-controversial fix.  The (3,8) knot looks fine parity-wise.
 * A metrically accurate picture of the (2,3)-torus knot would be good as well, I would think, so that people know there's a difference. The (3,8) picture looks like it was generated with r = 4 + cos(q\phi), your clarification about different parametrizations was helpful there.  --Vaughan Pratt (talk) 03:12, 14 September 2010 (UTC)

Selinger seems to be correct. Livingston ("Knot Theory", MAA Carus Monograph 24) shows a right-handed (3,5) and a left-handed (3,-5) torus knot on page 4. Besides, the braid word given in the Properties section applies to a (p,-q) (not (p,q)) torus knot, because the braid generators are left twists. The cause of the confusion seems to be the parametrisation: It generates a left-handed knot if p and q have the same sign. This should be fixed, even if Vaughan received a maths degree without caring about chirality. 77.3.111.217 (talk) 21:27, 24 October 2011 (UTC)
 * Selinger fixed the figures to his satisfaction on 14 September 2010. I don't have Livingston but if he's showing the opposite chirality from the current figure then one of Selinger or Livingston would appear to be physically incorrect.  (My mathematics degree only obligates to me to care that one chirality is not continuously deformable into the other, it is my physics degree that makes me care which is left and which is right in the figure. :) )  --Vaughan Pratt (talk) 13:09, 26 October 2011 (UTC)
 * I agree that this is merely about convention. But it would be preferable if those conventions matched those most common in the literature.  I have not seen many textbooks myself, many of which are out of print; please let me check two in my university library before you change anything.
 * Going by the history, Selinger did not change the figures or captions, they are the same as when they appeared initially. They had, and have, only left twists.
 * Regarding braids, the Wikipedia convention also seems to differ from the literature, where braid generators are apparently right twists (ex. http://www.math.unicaen.fr/~dehornoy/Books/Why/Dgr.pdf).   So the given braid word is correct if both chirality conventions are changed to what appears common.95.114.80.218 (talk) 18:28, 31 October 2011 (UTC)
 * Besides Livingston, only Murasugi (A survery of knot theory) had 3D drawings, which were right-handed for p,q>0. Lickorish (An introduction to knot theory) has drawings of the braids from which the torus can be constructed.  The braids are right-handed, giving a right-handed torus knot.  The other books I could find had no geometrical renderings.  So on balance I think the three figures (except the award, which looks right-handed to me) should have their sign of q reversed.  I am going to change that unless someone protests within the next week or so.Vswitchs (talk) 14:15, 5 November 2011 (UTC)
 * Correction: Murasugi's book is called "Knot theory and its applications". "A survey of knot theory" is by Akio Kawauchi and contains 2D drawing which are also right-handed for p,q>0.Vswitchs (talk) 14:57, 5 November 2011 (UTC)

The list section also says The figure on the right is torus link (72,4) at the top, but the image there claims to be a (36,3) torus link. --Belbury (talk) 09:30, 10 May 2023 (UTC)

Chirality question
It's clear that the alternative parametrization subtracting $$3\cos((p-q)\phi)$$ and $$3\sin((p-q)\phi)$$ from respectively x and y (while keeping $$r=\cos(q\phi)+4$$) in order to render torus knots more smoothly changes the parity of the knot in the (2,3) case, and I've also verified it for the (3,4) case. If it's true in general then it suffices to change the sign of z in order for (p,q) knots to be rendered smoothly with the same parity as for the defining parametrization. Is it? --Vaughan Pratt (talk) 01:51, 20 September 2010 (UTC)

On the other hand how much do we care about chirality here? The article knot theory has nothing of significance about it, other than to state that it is neglected when enumerating prime knots, i.e. a knot and its mirror image are considered equivalent. (For my senior (pure maths honours) thesis at Sydney University I picked knot theory, and never bothered with chirality which seemed a red herring.)

Torus knot definition not correct?
There seems to be an error in the definition of the torus knots. The (p,q)-torus knot is a curve that wraps the torus p times meridionally and q times longitudinally instead of what is written in the article (p times longitudinally and q times meridionally). So, the trefoil knot is the (3,2)-torus knot instead of the (2,3)-torus knot. At the moment, the definition used on wikipedia does not agree with the cited reference of Mathworld, and neither does it agree with the definition in The knot book: an elementary introduction to the mathematical theory of knots(Colin Conrad Adams). —Preceding unsigned comment added by 131.180.23.9 (talk) 15:25, 21 February 2011 (UTC)

The text is wrong. The 4th sentence should read: "The (p,q)-torus knot winds q times around a circle in the interior of the torus, and p times around its axis of symmetry." This becomes clear when the torus knot is cut open: p, not q, is the number of strands of the resulting braid. That said, (p,q) and (q,p) torus knots are probably equivalent in S3 at least, and possibly in R3 too? 77.3.111.217 (talk) 21:11, 24 October 2011 (UTC)
 * The fourth sentence of the first paragraph merely (correctly) describes the parametrization in the second paragraph, which explicitly specifies the embedding in R3 of the unit circle parametrized by $$\phi$$ (and implicitly that of the whole torus by replacing $$p\phi$$ by an independent parameter $$\theta$$). Any error therefore lies with the parametrization.  The third paragraph makes the point that "Other parametrizations are also possible, because knots are defined up to continuous deformation" and describes one way (out of many) to modify the second paragraph to agree with the figures.  Any discrepancy between the text and the figures is only possible with the third paragraph, so if you see one you'll need to point to where the discrepancy occurs in the third paragraph.
 * Regarding chirality, mathematics acknowledges that flipping an odd number of the xyz axes flips chirality, but is obliged to leave it to physics to distinguish left from right in the physical world of figures and string. Figures as we ordinarily understand them are physical, not mathematical.  --Vaughan Pratt (talk) 14:24, 26 October 2011 (UTC)
 * I may be confused... but I don't think the text is consistent with the parametrisation. As I understand the text, the knot is supposed to wrap p times around the circle defined by (x, y) = (2 cos φ, 2 sin φ).  But this wrapping takes the strand out of the x-y plane, changing its z coordinate, and the parametrisation has z=sin(qΦ) — q, not p.  Conversely, the text says the knot wraps q times around the torus' axis of symmetry, which is the z axis; but this rotation is within the x-y plane and in the parametrisation has the angle argument pΦ (not q).  The illustrations side with the parametrisation on this — they show (p, q) torus knots for p < q, and the winding number around the z axis (= the number of strands = the number of points in which the knot intersects any half plane extending from the z axis) is always the smaller number, p.  By cutting the knot open along such a half-plane, one obtains the braid mentioned farther down, which has p strands.  (I acknowledge that one could probably cut differently so as to obtain q strands, but that would be much less intuitive.)  — Preceding unsigned comment added by 95.114.80.218 (talk) 18:05, 31 October 2011 (UTC)
 * This seems to have been correct in the early versions of the page, until changed by Theraot on 10 June 2009. 95.114.80.218 (talk) 18:37, 31 October 2011 (UTC)
 * To answer the first two comments: The (p,q) and (q,p) torus knots are indeed equivalent. So it is not surprising that authors do not care too much on having consistent conventions and one will certainly find both definitions in the literature. -131.152.41.45 (talk) 13:50, 13 February 2012 (UTC)

list
here is a list of some example torus knots do you think i should put it in ? (i did'nt list trivial knots) {| class="wikitable" !p\q !3 !4 !5 !6 !7 !8 !9
 * 2
 * (3-2) torus knot.png
 * A (5,2)-torus knot.png
 * A (7,2)-torus knot.png
 * A (9,2)-torus knot.png
 * 3
 * A (4,3)-torus knot.png
 * A (5,3)-torus knot.png
 * A (7,3)-torus knot.png
 * A (8,3)-torus knot.png
 * 4
 * A (5,4)-torus knot.png
 * A (7,4)-torus knot.png
 * A (9,4)-torus knot.png
 * 5
 * A (6,5)-torus knot.png
 * A (7,5)-torus knot.png
 * A (8,5)-torus knot.png
 * A (9,5)-torus knot.png
 * 6
 * A (7,6)-torus knot.png
 * A (7,6)-torus knot.png
 * 7
 * A (8,7)-torus knot.png
 * A (9,7)-torus knot.png
 * 8
 * A (9,8)-torus knot.png
 * 5
 * A (6,5)-torus knot.png
 * A (7,5)-torus knot.png
 * A (8,5)-torus knot.png
 * A (9,5)-torus knot.png
 * 6
 * A (7,6)-torus knot.png
 * A (7,6)-torus knot.png
 * 7
 * A (8,7)-torus knot.png
 * A (9,7)-torus knot.png
 * 8
 * A (9,8)-torus knot.png
 * A (7,6)-torus knot.png
 * A (7,6)-torus knot.png
 * 7
 * A (8,7)-torus knot.png
 * A (9,7)-torus knot.png
 * 8
 * A (9,8)-torus knot.png
 * A (8,7)-torus knot.png
 * A (9,7)-torus knot.png
 * 8
 * A (9,8)-torus knot.png
 * A (8,7)-torus knot.png
 * A (9,7)-torus knot.png
 * 8
 * A (9,8)-torus knot.png
 * 8
 * A (9,8)-torus knot.png
 * A (9,8)-torus knot.png
 * A (9,8)-torus knot.png
 * A (9,8)-torus knot.png
 * A (9,8)-torus knot.png
 * A (9,8)-torus knot.png
 * A (9,8)-torus knot.png