Knot Atlas - User contributions [en]

8 19 Further Notes and Views

2020-12-18T02:57:14Z

AnonMoos:

[[8_19]] is the first non-homologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'off-diagonal' elements.)

{|
|- valign=top
{{Knot View Template|
image = 8-19-knotscape.png|
text = Knotscape}}
|
{{Knot View Template|
image = 8crossing-symmetrical-nonalternating.png|
text = Symmetrical form ; (3,4) torus knot}}
|
{{Knot View Template|
image = Trueloversknot.jpg|
text = True-lover's knot with sticked free ends }}
|
{{Knot View Template|
image = Pkamiejfilefkplj.png|
text = Equal to the previous, from knotilus }}
|- valign=top
{{Knot View Template|
image = 1,-2,3,4,-5,-1,2,-3,6,-7,8,5,-4,-6,7,-8.png|
text = Pretzel knot P(2,-3,-3) }}
|
{{Knot View Template|
image = logo-forumdelentrepreneuriat.png|
text = French logo }}
|
{{Knot View Template|
image = IMG 20180618 140044.jpg|
text = Seen in Singapore }}

|}

User talk:Drorbn

2018-08-06T04:07:57Z

AnonMoos: /* Perko comment on 10_32 */

==Thanks==
Hello again, Mr. Bar Natan. Thank you for helping me out with my problem. Dr. Conant said that your two examples were similar to his. Anyway, Dr. Conant told me that he became interested in knot theory because you showed up at a presentation he attended. I hope your knot theory career remains a successful one.

Strongbad, 2006-03-14 09:58-05:00

== Clarification? ==

Hi, I just e-mailed you about the "Borromean" bathroom tile, but then realized I could have more easily left a comment here...

Anyway, on the main page, you should probably make it clear that the Rolfsen table is for single-loop knots, while the Thistlewaite table is for multi-loop knots (for people who don't already know that in advance). Thanks. [[User:AnonMoos|AnonMoos]] 00:21, 27 Mar 2006 (EST)

:P.S., the "Shirt seen in Lisboa" at http://www.math.toronto.edu/~drorbn/Talks/Oporto-0407/KnotsInLisboa.html is actually a partial view of a monochromatic version of the U.S. Bicentennial emblem of 1976. See http://en.wikipedia.org/wiki/Image:Bicentlogo.png

==Borromean chain-mail knot?==
[[Image:Borromean-chain-mail.gif|thumb|center|150px]]
Which knot number is the "Borromean chain mail" knot? It's not L10a169, but I'm having difficulty determining which it actually is...
[[User:AnonMoos|AnonMoos]] 15:49, 27 Mar 2006 (EST)

Can't tell without a bit of a search, but I'm running out of time for today...
--[[User:Drorbn|Drorbn]] 17:42, 27 Mar 2006 (EST)

Ok, it is [[L10n107]].
--[[User:Drorbn|Drorbn]] 21:46, 27 Mar 2006 (EST)

:Ok, thanks (of course, I just assumed it was alternating without examining it, sorry). [[User:AnonMoos|AnonMoos]] 23:40, 28 Mar 2006 (EST)

==Linear decorative knot==

Sorry to keep bothering you, but I was looking at the simplest Celtic or pseudo-Celtic linear decorative knot, and it seems to be a real 8-crossing two-loop alternating link (when you shake it, it definitely does not fall apart), but I'm having difficulty relating it to any of the visual depictions on page [[The Thistlethwaite Link Table L8a1-L8a21]]... [[User:AnonMoos|AnonMoos]] 12:24, 2 Apr 2006 (EDT)

[[Image:Celtic-knot-simple.gif|300px|thumb|center]]

'''Cool!''' It is the mirror image of [[L8a8]], and it is not obvious to see that. In fact, I had to run the program [[KnotTheory`]] and see that the two have the same (i.e., opposite) invariants.

The Knot Atlas does not distinguish a knot from its mirror, so the picture should go on the [[L8a8]] page. --[[User:Drorbn|Drorbn]] 16:13, 2 Apr 2006 (EDT)

:Ok, thanks. It seems that there isn't really currently any program which will take a quasi-arbitrary input diagram and automatically report back "That's link #782 on the list." [[User:AnonMoos|AnonMoos]] 22:19, 3 Apr 2006 (EDT)

==Another linear decorative knot==

[[Image:Celtic-knot-simple-linear.gif|300px|thumb|center]]

I'm having difficulty in relating this to any of the list of alternating 10-crossing two-loop links, thanks... [[User:AnonMoos|AnonMoos]] 21:42, 20 Jun 2006 (EDT)

:Ouch! That was some strugle for me too, and it underlines the fact that my/our tools for doing such searches are not good enough. First I scanned the link table by hand and found nothing. Then I've quickly entered by hand a <tt>DTCode</tt> for your link, and it came out to be <tt>DTCode[{6, 12, 20, 16, 18}, {2, 4, 10, 8, 14}]</tt>. Then I computed the Jones polynomial of that and compared it with the Jones polynomials of all 10 crossings alternating links. Only two links differed from ours by a power of <math>q</math> - [[L10a31]] and [[L10a101]]. Of these the first gets ruled out immediately. The second seemed possible, but just to be sure I computed its Multivariable Alexander polynomial and found that it was complete different than the MVA of your link. After flipping the orientation of one of the components of your link so as to get <tt>DTCode[{6, 20, 12, 16, 14}, {2, 18, 8, 10, 4}]</tt>, everything is well.
:So the answer is [[L10a101]].
:--[[User:Drorbn|Drorbn]] 10:30, 23 Jun 2006 (EDT)

::Sorry if I was imposing an excessively fatiguing task on you, but I kind of like to correlate the decorative and/or symbolic motifs. Anyway, I'm already more or less approaching the limits of what has been done in that area with 11 or fewer crossings...
::I did download Knotscape, and booted Linux to run it, and that helped me identify [[:Image:Vodicka-3pointed-knot.gif]] as 10_75, but it seems that nothing except getting into the nitty gritty of the mathematics helps with multi-loop knots (links)...
::If two-loop links which are actually inherently symmetrical between the two loops (as L10a101 seems to be) were always portrayed symmetically, that would be helpful. [[User:AnonMoos|AnonMoos]] 22:16, 27 Jun 2006 (EDT)

==Here's one==

http://commons.wikimedia.org/wiki/Image:Gateknot.jpg
[[User:AnonMoos|AnonMoos]] 11:00, 6 Jul 2006 (EDT)

:Almost certainly it's either [[K11n34]] or [[K11n42]] and my hunch is that it's the former. It's a bit embarassing but the tools I have (and that are available through the Knot Atlas) cannot tell these two knot apart computationaly. So at the moment, to decide which of the two this is, one would have to draw and play with the drawings, or to tie a piece of string and play with it. --[[User:Drorbn|Drorbn]] 21:00, 11 Jul 2006 (EDT)

::Thanks... [[User:AnonMoos|AnonMoos]] 11:11, 15 Jul 2006 (EDT)

:::Things have now changed; see [[Heegaard Floer Knot Homology]].

==Me again==

I'm having a hard time mentally correlating [[:Image:Suchy-Vaud-Switz-COA.gif]] with [[L6n1]], but there don't seem to be a lot of other choices for non-alternating six-crossing three-loop links. If it comes down to it, I'm not absolutely 100% sure that [[:Image:Valknut-Symbol-3linkchain-closed.png]] is [[L6a5]] either (I thought so for a long time, but now that I look at all the six-crossing three-loop links side-by-side, my certainty is fading a little). Thanks... [[User:AnonMoos|AnonMoos]] 13:18, 20 Feb 2007 (EST)

:[[:Image:Suchy-Vaud-Switz-COA.gif]] does seem like [[L6n1]], to me. Take for example the upper ring (that is, the ring closest to the teeth of the key) in the "white link blue background" portion of the image, and spin it half way around a vertical axis in the plane of the screen. I believe you'll get [[L6n1]].
:It seems to me that [[:Image:Valknut-Symbol-3linkchain-closed.png]] is indeed mis-identified; it should be [[L6n1]] as well. --[[User:Drorbn|Drorbn]] 21:57, 22 Feb 2007 (EST)

::Thanks -- I moved to the L6n1 Further Notes and Views page accordingly. [[User:AnonMoos|AnonMoos]] 19:14, 24 Feb 2007 (EST)

==9 crossings?==
[[Image:3trefoil-9crossings.gif|right|250px]]

I'm having problems with this one -- it doesn't closely visually resemble any of the 9-crossing knots shown in the table, and all Knotscape will tell me is "Non-prime Dowker code". [[User:AnonMoos|AnonMoos]] 15:43, 2 Mar 2007 (EST)
:For indeed it is a non-prime knot - it is the "connected sum" of three [[3 1]]'s. Knots have a "factorization into primes" theorem, much like the factorization into primes theorem of number theory. So in some sense, "decomposable" knots are not interesting, because if you know all about prime knots you also know all about decomposable knots as well. Hence decomposable knots are not listed on the Knot Atlas. --[[User:Drorbn|Drorbn]] 16:03, 2 Mar 2007 (EST)
::OK, sorry if I'm embarrasingly ignorant of some of the basics, but I'm more interested in the decorative geometry aspects than in pure mathematical topology. Feel free to delete the image if it has no use here... [[User:AnonMoos|AnonMoos]] 18:31, 2 Mar 2007 (EST)
:::Why delete? It's nice to have, if only just on my talk page... --[[User:Drorbn|Drorbn]] 18:41, 2 Mar 2007 (EST)

[[Image:Celtic-insquare-three-fourths.gif|right|250px]]
----
Here's another visual variant from my efforts on decorative knots... Maybe these could be included on the 3_1 page? [[User:AnonMoos|AnonMoos]] 21:56, 4 May 2007 (EDT)
<br clear=all>

==(Hopefully) Final question on the decorative knots==
[[Image:Celtic-knot-basic-alternate.gif|200px|right]]
I was a little reluctant to ask you this before, since you were having such an arduous time identifying complex links, but if you could pin down this particular one (which comes in several decorative variations, and is the next step up from the symmetrical representation of [[L8a8]]), then it would pretty much complete the decorative knots series (at least with respect to knots and links with 11 or fewer crossings). This looks like it has 11 crossings, but I have reason to suspect that it may be reducible to a form with only 10 crossings..
:See [[The_Multivariable_Alexander_Polynomial#Detecting_a_Link_Using_the_Multivariable_Alexander_Polynomial]]. --[[User:Drorbn|Drorbn]] 16:16, 3 May 2007 (EDT)
::Thanks (hope it wasn't too strenuous an effort!). I only played with Mathematica briefly a number of years ago, and don't have access to it now... [[User:AnonMoos|AnonMoos]] 21:58, 4 May 2007 (EDT)

==problems==
The "Data:XXX/KhovanovTable" inclusions seem to be broken, and the software refuses to generate thumbnails for the image [[:Image:Lord_Boyce_Cinque_Ports_badge.gif]] I uploaded... [[User:AnonMoos|AnonMoos]] 13:00, 11 March 2009 (EDT)

:I uploaded a PNG, [[:Image:Lord_Boyce_Cinque_Ports_badge.png]] and it kind of worked (thumbnails were generated), but the thumbnails are in 16-bit per channel format, which means that they're larger than they need to be, and may not be handled by some software programs. If you updated the Wikimedia software, it seems like you should probably configure some of the graphics parameters of the new version... [[User:AnonMoos|AnonMoos]] 10:32, 2 April 2009 (EDT)

==Further annoying questions==
I can't really tell from visual inspection whether image [[:Image:10crossing-2trefoil.png]] falls under [[10 165]], and unfortunately, I really don't understand the Dowker-code generating process well enough to boil [[:Image:10crossing-2trefoil.png]] down to a sequence of 10 numbers (I always get a sequence of 20 numbers), so I can't use Knotscape to check. Also, going through my uploaded images, I guess that [[:Image:Bar-knot-simple-decorative.gif]] from 2006 was never classified. Thanks... [[User:AnonMoos|AnonMoos]] 09:13, 4 February 2010 (EST)

:See http://katlas.math.toronto.edu/drorbn/AcademicPensieve/2010-02/nb/KnotsFromAnonMoos.pdf. [[User:Drorbn|Drorbn]] 07:23, 7 February 2010 (EST)

::Thanks, but the instructions on article [[DT (Dowker-Thistlethwaite) Codes]] are not really specific and detailed enough for me to do exactly what you did on the first page of the PDF file by following them, and I'm pretty sure I followed a kind of simplified version of the procedure when diagnosing [[:Image:Vodicka-3pointed-knot.gif]] as [[10 75]], my main previous Dowker effort (hope that doesn't mean the identification is incorrect!). It might have been easier for me to visually spot 10_120 amid the list of 10-crossing knots if the images didn't sometimes have an annoying tendency to depict symmetrical knots in a unnecessarily visually asymmetric way... [[User:AnonMoos|AnonMoos]] 10:19, 7 February 2010 (EST)

:::Yes, you have [[10 75]] right. See http://katlas.math.toronto.edu/drorbn/AcademicPensieve/2009-06/one/Question_from_Jorgen.pdf for a time when I was asked about it by somebody else. As for the pictures, the ones of up to 10 crossings were made by Rob Scharein of [http://www.knotplot.com/ KnotPlot] to resemble as much as possible the images in Rolfsen's book. The ones for 11 crossings were generated in bulk by a program written by Thistlethwaite; things generated in bulk would never be perfect. [[User:Drorbn|Drorbn]] 06:43, 8 February 2010 (EST)

Well anyway, I forgot about the "Draw-a-knot" or LinkSmith feature of Knotscape; I used that to identify [[:Image:9crossings-knot-symmetrical.png]] with [[9_40]] just recently (though it's also somewhat laborious to use). [[User:AnonMoos|AnonMoos]] 16:20, 12 February 2010 (EST)

==Template editing==
I really don't understand such template editing. [[User:AnonMoos|AnonMoos]] 10:50, 25 February 2010 (EST)

==Further small details==
I created page [[Notes_on_presentations_of_10_60]] by mistake, should be deleted. Also, I couldn't help noticing that Image:10_60.gif is unfortunately rather poor (you have to stare at it for a while to even figure out what's a structural not crossing and what isn't...) -- [[User:AnonMoos|AnonMoos]] 19:02, 27 May 2010 (EDT)

==Bogus page creations==
I accidentally created [[Notes on presentations of K13a4532]] (could be deleted) and someone else created [[Notes on presentations of 9 47]] when I'm not too certain that's what he actually intended to do (his [[Special:Contributions/Knotopologynn|previous contributions of the same type]] were all to "Further Notes and Views" pages, not "Notes on presentations" pages...) -- [[User:AnonMoos|AnonMoos]] 10:49, 17 January 2011 (EST)

==Are these two the same?==
[[Image:Non-Borromean-rings minimal-overlap.png|left|150px]]
[[Image:Non-Borromean-rings minimal-overlap2.png|right|150px]]
We've been kind of assuming they are since February 2007 (above), but it would be nice to know for sure... [[User:AnonMoos|AnonMoos]] 07:31, 3 July 2011 (EDT)

:Never mind, I found the info here: http://www.liv.ac.uk/~spmr02/rings/types.html ... -- [[User:AnonMoos|AnonMoos]] 14:16, 3 July 2011 (EDT)

==[[Help talk:Usage]]==
I've got a question at [[Help talk:Usage#Search/Find]]. Thanks. [[User:Hyacinth|Hyacinth]] 19:04, 1 May 2013 (EDT)

==Khovanov Homology==
For a lot of links, this seems to be displayed as raw HTML or a link to a non-existent page... [[User:AnonMoos|AnonMoos]] 06:00, 2 May 2013 (EDT)

==Image resizing==
For a while there was no thumbnailing of newly-uploaded GIFs, but now most or all GIFs and PNGs seem to be having thumbnailing problems. [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 03:38, 21 March 2016 (EDT)

:For example, look at page [[L6n1]]. A number of images display very roughly in my browser, a sign that the browser is dropping rows and columns from the full-size image on the fly, a crude form of resizing which gives poorer results than true image thumbnailing. So if I right click on the "Basic symmetrical depiction" image and select "Copy image location", I get <tt>http://katlas.math.toronto.edu/w/images/e/ec/Non-Borromean-rings_minimal-overlap2.png</tt>, which is a full-sized image which gets crudely squashed down by the end-user's browser. A link for a thumbnail created by Wikimedia software looks more like <tt>https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Two-representations-of-L6n1-link-as-linked-circles.svg/120px-Two-representations-of-L6n1-link-as-linked-circles.svg.png</tt> -- notice the image name appears twice, and the "120px". I'm not sure that any image thumbnailing on this site is working at all. [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 11:04, 7 May 2016 (EDT)

==You have a spammer==
Should not have approved NealWrang, it appears... [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 20:24, 12 July 2016 (EDT)

Not any more... --[[User:Drorbn|Drorbn]] ([[User talk:Drorbn|talk]]) 12:35, 13 July 2016 (EDT)

==Perko comment on 10_32==
Perko seems to have left a comment about [[10 32]] at https://en.wikipedia.org/wiki/Talk:The_Knot_Atlas ... -- [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 00:07, 6 August 2018 (EDT)

User talk:Robert FERREOL

2018-08-06T03:53:19Z

AnonMoos: /* 8 18 */

Dear Robert,

Thanks for the nice photo of [[9_40]]! Can you add a note there saying in just a word or two where the picture was taken?

Best,

--[[User:Drorbn|Drorbn]] 20:53, 4 August 2007 (EDT)

== Sep 1, 2015 ==

Robert -

By a classical theorem, an alternating link without "nugatory" crossings cannot be simplified. Hence the image you added, [[File:Noeudmongol.jpg|50px]], is an image of a 12-crossing link and cannot be an image of [[L8a14]].

Best,

Dror.

obviously, this is a big mistake

robert

==[[8 18]]==

"8 18" is an alternating knot, while [[:File:IMG 20180618 140044.jpg]] is non-alternating. [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 23:48, 5 August 2018 (EDT)

:It's probably [[8_19]]... [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 23:53, 5 August 2018 (EDT)

User talk:Robert FERREOL

2018-08-06T03:48:22Z

AnonMoos: 8 18

8 18 Further Notes and Views

2018-08-06T03:45:35Z

AnonMoos: rv to 7:13, 26 September 2016

<noinclude>Back to [[8_18]].</noinclude>
<br>
{|
|- valign=top
{{Knot View Template|
image = IGKT_h120.jpg |
text = Logo of the International Guild of Knot Tyers [http://www.igkt.net/]}}
|
{{Knot View Template|
image = ABeneficenciaFamiliar_120.jpg |
text = A charity logo in Porto [http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/ABeneficenciaFamiliar.html]}}
|
{{Knot View Template|
image = LaserCut_8_18_120.jpg |
text = A laser cut by Tom Longtin [http://mysite.verizon.net/t.longtin/knot_atlas/index.html]}}
|
{{Knot View Template|
image = Bar-knot-simplest-decorative.gif |
text = Knot in (pseudo-)Celtic decorative form}}
|- valign=top
{{Knot View Template|
image = 8crossings-4circles.png |
text = Less symmetrical}}
|
{{Knot View Template|
image = 8crossing-circular.png |
text = Within outer circle}}

|
{{Knot View Template|
image = noeudcarre.png |
text = Impossible figure}}
|
{{Knot View Template|
image = mongolian8.18.gif |
text = Mongolian ornament}}

|- valign=top
{{Knot View Template|
image = Carrick mat by Brianetta.jpg |
text = Jump rope knot}}
|
{{Knot View Template|
image = Belt-design.jpg |
text = Belt design}}
|
{{Knot View Template|
image = Bondage_knot.jpg |
text = Bondage knot}}
|
{{Knot View Template|
image = fhdebifd.png |
text = Spheric depiction}}

|- valign=top
{{Knot View Template|
image = Képi Infanterie Sous-Lieutenant Armée Française.jpg|
text= A "Hungarian Knot", decorating French Military uniforms.
}}
|}

L8a14 Further Notes and Views

2018-08-06T03:36:20Z

AnonMoos:

<noinclude>Back to [[L8a14]].</noinclude>
{|
|- valign=top
{{Knot View Template|
image = StockExchangePalace_160.jpg |
text = Floor of the Stock Exchange [http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/StockExchangePalace.html]}}
|
{{Knot View Template|
image = Star-lakshmi.gif |
text = Represented as two interlaced squares}}
|
{{Knot View Template|
image = IstanbulFloorMosaic.jpg |
text = A mosaic seen on an Istanbul floor}}
|
{{Knot View Template|
image = Jaromer-Czech-CoA.png |
text = Coat of arms of Jaroměř, Czech Republic}}
|- valign=top
{{Knot View Template|
image = St-Savior-Jersey-UK-COA.gif |
text = Coat of arms of St. Savior, Jersey, Channel Islands, depicting Crown of Thorns religious symbol}}
|
{{Knot View Template|
image = Sameba Cathedral Tbilisi 13.jpg |
text = Cathedral in Tbilisi, Georgia}}
|
{{Knot View Template|
image = Mozota-Coatofarms.gif |
text = Coat of arms of Mozota, Spain}}
|
{{Knot View Template|
image = Decorative motif of interlaced squares, San Pancrazio, Florence.jpg |
text = Decorative motif of interlaced squares, San Pancrazio, Florence}}
|- valign=top
{{Knot View Template|
image = Macedonian cross.png |
text = Macedonian cross}}
|
{{Knot View Template|
image = 8crossings-link-3D.PNG |
text = 3D depiction}}
|
{{Knot View Template|
image = Visitacao Braga Brazao.jpg |
text = ''Arma Christi'' carving in monastery in Portugal}}
|
{{Knot View Template|
image = Handewitt Schleswig-Holstein arms oak.png |
text = Handewitt, Schleswig-Holstein oak leaves and acorns}}
|
|- valign=top
{{Knot View Template|
image = Onam festival flower carpet Maspraveen.jpg |
text = Flower carpet, south India}}
|
{{Knot View Template|
image = Flower carpet onam . irvin 01.jpg |
text = Flower carpet, south India}}
|
{{Knot View Template|
image = Persian or central Asian tile detail 15th century.JPG |
text = Islamic art tile}}

|
{{Knot View Template|
image = 8crossingsmarocco.jpg |
text = moroccan craft}}
|- valign=top
{{Knot View Template|
image = Murato.jpg |
text = Link on a church of Murato,Corsica }}

|}

File:2017-12-07 10.44.34.jpg

2018-01-25T03:52:49Z

AnonMoos:

Mexican book.

<P>See https://commons.wikimedia.org/wiki/File:Triquetra-circle-interlaced.svg for similar...

L9a33 Further Notes and Views

2018-01-25T03:49:41Z

AnonMoos:

<noinclude>Back to [[L9a33]].</noinclude>
<br>
{|
|- valign=top
{{Knot View Template|
image = L9a33-symm-circ-arcs.png|
text = Symmetric form}}
|
{{Knot View Template|
image = L9a33-symm-circ-arcs-alt.png|
text = Alternate symmetric version, with three lines touching at center}}
|
{{Knot View Template|
image = L9a33-symm-circ-arcs-7enclosed.png|
text = Alternate symmetric version, with three lines touching at circumference}}
|- valign=top
{{Knot View Template|
image = Link9-corners.png |
text = Form made from 45-degree lines and circular arcs.}}
|
{{Knot View Template|
image = Acdeicjb.png |
text = Depiction obtained by knotilus.}}
|
{{Knot View Template|
image = Oddcnlclmfplejdg.png |
text = With an hypotrochoid [http://www.mathcurve.com/courbes2d/hypotrochoid/hypotrochoid.shtml].}}
|- valign=top
{{Knot View Template|
image = 2017-12-07_10.44.34.jpg|
text = Mexican book.}}

|}

L8a13 Further Notes and Views

2017-02-21T02:13:46Z

AnonMoos:

<noinclude>Back to [[L8a13]].</noinclude>
<br>
{|
|-
{{Knot View Template|
image = L8a13-symm.png|
text = Symmetric form}}
|
{{Knot View Template|
image = Solomons-two.png|
text = (alternate)}}
|
{{Knot View Template|
image = 2017-02-14 11. zurich.jpg|
text = [https://www.nationalmuseum.ch/e/ National swiss museum]}}

|
{{Knot View Template|
image = gkmenihdgfpgioch.png|
text = Povray depiction}}

|}

L9a40 Further Notes and Views

2017-02-21T02:01:13Z

AnonMoos:

<noinclude>Back to [[L9a40]].</noinclude>
{|
|-

{{Knot View Template|
image = zurich2.jpg|
text = [https://www.nationalmuseum.ch/e/ National swiss museum]}}

{{Knot View Template|
image =
Bnokfdccoamjjhjc.png|
text = Povray depiction}}
|}

L9a32 Further Notes and Views

2017-02-06T03:08:01Z

AnonMoos:

<noinclude>Back to [[L9a32]]</noinclude>
{| style="background: transparent;"
|- valign=top
{{Knot View Template|
image = Triquetra-Interlaced-Triangle-Circle.png |
text = A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")
}}
|
{{Knot View Template|
image = Triquetra-circle-interlaced-smootharcjoins.png |
text = Symmetric version, with three lines touching at center
}}
|
{{Knot View Template|
image = Triquetra-circle-interlaced-7enclosed.png |
text = Symmetric version, with three lines touching at cicrumference
}}
|-
{{Knot View Template|
image = CUMC.JPG |
text = Logo of the [http://cumc.math.ca Canadian Undergraduate Mathematics Conference]
}}
|
{{Knot View Template|
image = ndkkoebhdfnbkdcf.png|
text = [http://www.mathcurve.com/courbes2d/hypotrochoid/hypotrochoid.shtml Circled hypotrochoid ]
}}
|}

L9a33 Further Notes and Views

2017-02-06T03:05:00Z

AnonMoos:

L9a33 Further Notes and Views

2017-02-06T03:04:38Z

AnonMoos:

<noinclude>Back to [[L9a33]].</noinclude>
<br>
{|
|- valign=top
{{Knot View Template|
image = L9a33-symm-circ-arcs.png|
text = Symmetric form}}
|
{{Knot View Template|
image = L9a33-symm-circ-arcs-alt.png|
text = Alternate symmetric version, with three lines touching at center}}
|
{{Knot View Template|
image = L9a33-symm-circ-arcs-7enclosed.png|
text = Alternate symmetric version, with three lines touching at circumference}}
|- valign=top
{{Knot View Template|
image = Link9-corners.png |
text = Form made from 45-degree lines and circular arcs.}}
|
{{Knot View Template|
image = Acdeicjb.png |
text = Depiction obtained by knotilus.}}

{{Knot View Template|
image = Oddcnlclmfplejdg.png |
text = With an hypotrochoid [http://www.mathcurve.com/courbes2d/hypotrochoid/hypotrochoid.shtml].}}

|}

Ornamental depictions of links with 12 crossings

2017-02-06T02:59:40Z

AnonMoos:

Some decorative depictions of links with 12 crossings :

;With two components

;a) Components are loops
<gallery>
Image:Two-interlaced-hexagrams.png|Two interlaced open hexagrams
Image:Ccross-two-link.png|Quasi-Celtic two-loop link
Image:12crossing-twosymmetric-link.png|Two loops
Image:Link12-round.png|Two loops
Image:Three-L4a1.png|Two-loop link
Image:Four-loop-interlaced-circle.png|Four-fold looping interlaced with circle (two-loop link)
Image:Noeudmongol.jpg|Mongolian ornament, equivalent to preceding
Image:Pleter50.png|Equivalent to preceding
Image:12-crossing-link Tallara Lousame Galicia Spain.jpg|Sculpture of preceding
Image:Carreau.jpg|Tiling ; equivalent to preceding
Image:assiette.sicile.jpg|Sicilian plate, also equivalent
Image:12crossings-pseudo-Celtic-link.png|One crossing of [[K13a1345]] eliminated, converting from knot to link...
</gallery>

;b) One component is knoted
<gallery>
Image:Triquetra-double-interlaced.png|Two mutually-interlaced triquetras
Image:Trefoil-in-trefoil.png|Trefoil interlinked with smaller trefoil
Image:Two-trefoils-triangles-link.png|Two interlinked trefoils
Image:12crossing-link-inrectangle.png|Two interlinked trefoils inside rectangle
Image:Two-trefoils-granny-12crossings.png|Sum of two trefoils linked with a loop
</gallery>

;With three components
<gallery>
Image:Celtic-knot-square-3loops.png|3-loop link in the form of a quasi-Celtic knot
Image:Cross of Uppland Runestone-.jpg|Ca. 1000 A.D. runestone, equivalent to preceding
Image:12-crossings-squares-16.png|Equivalent link in square form
Image:Three-loop-link-12-crossings-alternate.png|(alternate version)
Image:Brunnian-3-not-Borromean.gif|Three-component [[Brunnian link]]
Image:Brunnian-link-12crossings-nonBorromean-quasi-Arabesque.png|More ornate Brunnian link depiction
Image:Three-L4a1-config.png|Three-loop link with three [[L4a1]] configurations
Image:12crossings-pseudo-Borromean.png|Three-loop link
Image:Triang-two-circ.png|Triangle and two circles
Image:12crossings-threesymmetric-link.png|Three loops symmetric
Image:12crossings-decorative-threefold-incircle-link.png|Three loops symmetric
Image:Two-trefoils-loop-12crossings.png|Two trefoils and one loop
</gallery>

;With four loops
<gallery>
Image:4squares-12crossings.png|Four linked squares
Image:4intersecting-rounded-rectangles.png|Variant
Image:fourcircles.jpg|Variant with circles
Image:noeudsCarres-2.png|Variant with rectangles as an impossible object
Image:adiadgib.jpg|Variant on the sphere ; the crossings are the vertices of the cuboctahedron
Image:Borromean-plus-triangle.png|Four loops (Borromean + triangle)
</gallery>

;With six loops
<gallery>
Image:Six-rings.png|Six rings
Image:Six-link-closed-chain-Hisham-Umayyad-palace-Jericho.jpg|Decoration equivalent to a circle of six interlinked loops
</gallery>

For ornamental depictions of knots with 12 crossings, see [[K12a477]], [[K12a503]], [[K12a541]], [[K12a561]], [[K12a975]], [[K12a991]], [[K12a1019]], [[K12a1210]], [[K12n242]], and (non-prime) [[:Image:Trefoil-of-trefoils.png]], [[:Image:Celtic-knot-insquare-green-transparentbg.png]], [[:Image:Three-figure8-knot triang1.png]], and [[:Image:Three-figure8-knot triang2.png]].

Talk:8 19 Further Notes and Views

2017-01-19T15:35:21Z

AnonMoos: Created page with "The illustration at http://www.npr.org/sections/thetwo-way/2017/01/12/509353074/scientists-have-twisted-molecules-into-the-tightest-knot-ever looks a lot like :File:8crossin..."

The illustration at http://www.npr.org/sections/thetwo-way/2017/01/12/509353074/scientists-have-twisted-molecules-into-the-tightest-knot-ever looks a lot like [[:File:8crossing-symmetrical-nonalternating.png]]... -- [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 10:35, 19 January 2017 (EST)

User talk:BrianGilbert

2016-11-15T16:23:55Z

AnonMoos: Created page with "==Ideal knots== The link http://fizyka.phys.put.poznan.pl/~pieransk/TablesUpTo9.html no longer works. Also, it might be nice to give a visual representation (image) for a..."

==[[Ideal knots]]==
The link http://fizyka.phys.put.poznan.pl/~pieransk/TablesUpTo9.html no longer works. Also, it might be nice to give a visual representation (image) for at least a few cases... [[User:AnonMoos|AnonMoos]] ([[User talk:AnonMoos|talk]]) 11:23, 15 November 2016 (EST)

User talk:Drorbn

2016-07-13T00:24:52Z

AnonMoos: /* You have a spammer */

@Helper Quickbooks 1 800 728 7356 pro support phone number && Quickbooks tech support phone number

2016-07-13T00:20:57Z

AnonMoos: AnonMoos moved page @Helper Quickbooks 1 800 728 7356 pro support phone number && Quickbooks tech support phone number to SpamJunk1: spam junk

#REDIRECT [[SpamJunk1]]

User talk:Drorbn

2016-05-07T15:04:50Z

AnonMoos: /* Image resizing */

User talk:Drorbn

2016-03-21T07:38:44Z

AnonMoos: /* Image resizing */

File:8-4 Knot.png

2016-03-21T07:27:33Z

AnonMoos: AnonMoos uploaded a new version of "File:8-4 Knot.png": technical image cleanup

Somewhat symmetric representation of [[8_4]] knot, made from circular arcs and 45° straight lines.

Created by AnonMoos, declared to be in the public domain.

Generated from the following vector PostScript source code:
<pre>%!
306 496 translate .4 dup scale 32 setlinewidth .92 .96 1 setrgbcolor
1 setlinejoin/z{20000 sqrt}def/y{gsave 60 setlinewidth 0 setgray}def
z 2 mul z -2 mul 100 315 45 arc 0 0 100 45 135 arc
z -2 mul z -2 mul 100 135 225 arc z z -5 mul 100 225 45 arc
z -2 mul 0 100 225 45 arcn 0 z neg lineto
z 2 mul 0 100 135 315 arcn z neg z -5 mul 100 135 315 arc
closepath y stroke grestore stroke 0 setlinejoin
y 16 z add 16 moveto -32 -32 rlineto stroke
16 z sub 16 moveto -32 -32 rlineto stroke
16 16 z 3 mul sub moveto -32 -32 rlineto stroke
16 16 z 5 mul sub moveto -32 -32 rlineto stroke
-16 z 2 mul add 16 z sub moveto 32 -32 rlineto stroke
-16 z 2 mul sub 16 z sub moveto 32 -32 rlineto stroke
-16 z add 16 z 4 mul sub moveto 32 -32 rlineto stroke
-16 z sub 16 z 4 mul sub moveto 32 -32 rlineto stroke
grestore 18 z add 18 moveto -36 -36 rlineto stroke
18 z sub 18 moveto -36 -36 rlineto stroke
18 18 z 3 mul sub moveto -36 -36 rlineto stroke
18 18 z 5 mul sub moveto -36 -36 rlineto stroke
-18 z 2 mul add 18 z sub moveto 36 -36 rlineto stroke
-18 z 2 mul sub 18 z sub moveto 36 -36 rlineto stroke
-18 z add 18 z 4 mul sub moveto 36 -36 rlineto stroke
-18 z sub 18 z 4 mul sub moveto 36 -36 rlineto stroke showpage
%EOF</pre>

6 3 Further Notes and Views

2016-03-18T02:48:32Z

AnonMoos:

<noinclude>Back to [[6_3]]</noinclude>
{|
{{Knot View Template|
image = Blue_6_3_Knot.png |
text = 3D depiction}}
|
{{Knot View Template|
image = Noeudirlandais.jpeg |
text = Irish knot, sum of four 6.3}}
|}

7 4 Further Notes and Views

2016-03-18T02:45:02Z

AnonMoos:

<noinclude>Back to [[7_4]].</noinclude>
{|
|- valign=top
{{Knot View Template|
image = Celtic-knot-linear-7crossings.png |
text = Celtic or pseudo-Celtic knot}}
{{Knot View Template|
image = mongolian74.gif |
text = Mongolian ornament}}
{{Knot View Template|
image = SusanWilliamsMedallion_160.jpg |
text = Susan Williams' medallion [http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/SusanWilliamsMedallion.html], the "Endless knot" of Buddhism [http://en.wikipedia.org/wiki/Endless_knot]}}
|- valign=top
{{Knot View Template|
image = Endless knot outlined.png |
text = Ornamental "Endless knot"}}
{{Knot View Template|
image = KornikKnot_160.jpg |
text = a knot seen at the Castle of Kornik [http://www.math.toronto.edu/~drorbn/Gallery/KnottedObjects/KornikKnot.html]}}
{{Knot View Template|
image = Takara-4harts-C7.jpg |
text = A 7-4 knot reduced from TakaraMusubi with 9 crossings
[http://intervision.aadau.net/]}}
|- valign=top
{{Knot View Template|
image = TakaraMusubi-C9.jpg |
text = TakaraMusubi knot seen in Japanese symbols, or Kolam in South India
[http://intervision.aadau.net/]}}
{{Knot View Template|
image = Ashtamangala-Endless-knot.jpg |
text = Buddhist Endless Knot}}
{{Knot View Template|
image = Endless-knot-Shrivatsa.jpg |
text = Ornamental Endless Knot}}
|- valign=top
{{Knot View Template|
image = Duerer-knot.jpg |
text = Albrecht Dürer knot, 16th-century}}
{{Knot View Template|
image = LaserCut_7_4_120.jpg |
text = A laser cut by Tom Longtin [http://mysite.verizon.net/t.longtin/knot_atlas/index.html]}}
{{Knot View Template|
image = Interwoven unicursal hexagram.gif |
text = Unicursal hexagram of occultism}}
|- valign=top
{{Knot View Template|
image = p11.gif |
text = Lissajous curve : x=cos 3t , y=sin 2t, z=sin 7t}}
|}

L6a3 Further Notes and Views

2016-03-18T02:42:44Z

AnonMoos:

<noinclude>Back to [[L6a3]]</noinclude><br>
{|
|- valign=top
{{Knot View Template|
image = 6a3Oberwolfach92_160.jpg |
text = Ruberman, Cochran, Melvin, Akbulut, Gompf, Kirby [http://www.maths.warwick.ac.uk/gt/Rob_Kirby_etc/6-96.html]}}
|
{{Knot View Template|
image = 6a3BySchwartz_120.png |
text = Rich Schwartz' "72" [http://www.math.umd.edu/~res/poster.html]}}
|
{{Knot View Template|
image = Triangle-Interlaced-Circle.gif |
text = Triangle interlaced with a circle, a traditional symbol of the Christian Trinity (less used in recent centuries) }}
|
{{Knot View Template|
image = Trefoil-Architectural-Equilateral-Triangle-interlaced.png |
text = An architectural trefoil (the outline of three overlapping circles) interlaced with an equilateral triangle, another old Christian Trinitarian symbol.}}
|}
<div class="NavFrame"><div class="NavHead">Further images</div>
<div class="NavContent">
{| style="background: transparent;"
|- valign=top
{{Knot View Template|
image = Celtic flag proposed.png |
text = Celtic flag proposal.}}
{{Knot View Template|
image = Blonay-Vaud-Switzerland-flag.gif |
text = Flag of Blonay, Vaud, Switzerland.}}
{{Knot View Template|
image = Eye of Providence inside triangle interlaced with circle Venice.jpg |
text = "Eye of Providence" inside triangle interlaced with circle, Venice.}}
{{Knot View Template|
image = Six-circles-star-David.png |
text = Composed of intersecting circles.}}
|-
{{Knot View Template|
image = Hexagramm aus Nepal.jpg |
text = Hindu hexagram symbol with Aum/Om, from Nepal.}}
{{Knot View Template|
image = double boomerang.jpg |
text = Double boomerang-like Kolam and Linked knot, Nagata.}}
{{Knot View Template|
image = carpet612.jpg |
text = A carpet.}}
|}
</div></div>
<br clear=left>

L6a1 Further Notes and Views

2016-03-18T02:38:44Z

AnonMoos:

<noinclude>Back to [[L6a1]].</noinclude>
{|
|- valign=top
{{Knot View Template|
image = Link-L6a1-C.jpg|
text = A kolam with two cycles/components[http://www.scipress.org/journals/forma/frame/22.html]}}
{{Knot View Template|
image = Eegchccb.png|
text = Depiction with two eights interlaced}}
{{Knot View Template|
image = mongolian632.gif|
text = Mongolian ornament ; the two eights are horizontal}}
|- valign=top
{{Knot View Template|
image = MongolianL6a1.gif|
text = Another one, sum of two L6a1}}
{{Knot View Template|
image = 1,2,3,1,4,5,6,4-3,5,6,2.png|
text = Another depiction}}
|}

10 116 Further Notes and Views

2016-03-18T02:36:13Z

AnonMoos:

<noinclude>Back to [[10 116]]</noinclude><br>

{|
|- valign=top
{{Knot View Template|
image = 10crossings square.png |
text = Square depiction.}}
|
{{Knot View Template|
image = Triquetra-heart-knot.png |
text = Triquetra-heart decorative depiction.}}
|
{{Knot View Template|
image = Knot_manuscript_NKP_XXIII_F_138_02r.jpg |
text = Medieval manuscript.}}
|- valign=top
{{Knot View Template|
image = Double-heart-knot 10crossings.png |
text = As decorative double heart knot.}}
|
{{Knot View Template|
image = Mongolian10.116.gif |
text = Mongolian ornament.}}
|}

L9a33 Further Notes and Views

2016-03-18T02:33:13Z

AnonMoos:

8 18 Further Notes and Views

2016-03-18T02:27:34Z

AnonMoos:

6 1 Further Notes and Views

2016-03-18T02:21:39Z

AnonMoos:

<noinclude>Back to [[6_1]]</noinclude><br>
{|
{{Knot View Template|
image = StevedoresKnot-1-6a-C.jpg|
text = A Kolam of a 3x3 dot array
}}
{{Knot View Template|
image = Blue Stevedore Knot.png|
text = 3D depiction
}}

{{Knot View Template|
image = Stevedor.png|
text = Polygonal depiction
}}
|-
{{Knot View Template|
image = squarestevedore.png|
text = Simple square depiction
}}

{{Knot View Template|
image = stevedore2.png|
text = An other one
}}
{{Knot View Template|
image = necklace.jpg|
text = Necklace
}}
|}

9 40 Further Notes and Views

2016-03-18T02:16:56Z

AnonMoos:

<noinclude>Back to [[9_40]].</noinclude>
{|
|- valign=top

{{Knot View Template|
image = 9crossings-knot-symmetrical.png|
text = In three-fold symmetrical form}}
|
{{Knot View Template|
image = 9crossings-knot-symmetric-triangles-pseudo-valknut.png|
text = Symmetrical triangular form}}
|
{{Knot View Template|
image = 9crossings-knot-symmetric-triangles-quasi-valknut.png|
text = (less open)}}
|
{{Knot View Template|
image = 9crossings-knot-symmetric-triangles-quasi-valknut-alternate.png|
text = (alternate)}}

|- valign=top
{{Knot View Template|
image = noeud triangle.png|
text = Variant}}
|
{{Knot View Template|
image = Chaise_alsacienne.jpg|
text = Photo of an alsatian chair, France.}}
|
{{Knot View Template|
image = cylindric9.40.png|
text = Cylindrical depiction.}}
|}

9 40 Further Notes and Views

2016-03-18T02:15:48Z

AnonMoos:

<noinclude>Back to [[9_40]].</noinclude>
{|
|- valign=top

{{Knot View Template|
image = 9crossings-knot-symmetrical.png|
text = In three-fold symmetrical form}}
|
{{Knot View Template|
image = 9crossings-knot-symmetric-triangles-pseudo-valknut.png|
text = Symmetrical triangular form}}
{{Knot View Template|
image = 9crossings-knot-symmetric-triangles-quasi-valknut.png|
text = (less open)}}
|
{{Knot View Template|
image = 9crossings-knot-symmetric-triangles-quasi-valknut-alternate.png|
text = (alternate)}}

|- valign=top
{{Knot View Template|
image = noeud triangle.png|
text = Variant}}
|
{{Knot View Template|
image = Chaise_alsacienne.jpg|
text = Photo of an alsatian chair, France.}}
|
{{Knot View Template|
image = cylindric9.40.png|
text = Cylindrical depiction.}}
|}

8 4 Further Notes and Views

2016-03-18T02:01:54Z

AnonMoos:

<noinclude>Back to [[8_4]]</noinclude><br>
{|
{{Knot View Template|
image = 8-4 Knot.png|
text = Somewhat symmetric representation
}}
|}

8 4 Further Notes and Views

2016-03-18T01:59:53Z

AnonMoos: Created page with "<noinclude>Back to 8_4</noinclude><br> {{Knot View Template| image = 8-4 Knot.png| text = Somewhat symmetric representation }}"

<noinclude>Back to [[8_4]]</noinclude><br>

{{Knot View Template|
image = 8-4 Knot.png|
text = Somewhat symmetric representation
}}

File:8-4 Knot.png

2016-03-18T01:56:06Z

AnonMoos: Somewhat symmetric representation of 8_4 knot, made from circular arcs and 45° straight lines. Created by AnonMoos, declared to be in the public domain. Generated from the following vector PostScript source code: <pre>%! 306 496 translate .4 dup...

File:8crossing-symmetrical-nonalternating.png

2013-11-30T11:48:27Z

AnonMoos:

Self-made graphic, declared to be in public domain, see http://commons.wikimedia.org/wiki/File:8crossing-symmetrical-nonalternating.svg

File:8crossing-symmetrical-nonalternating.png

2013-11-30T11:48:10Z

AnonMoos:

Self-made graphic, declared to be in public domain, see https://commons.wikimedia.org/wiki/File:8crossing-symmetrical-nonalternating.svg

8 19 Further Notes and Views

2013-11-30T11:16:47Z

AnonMoos:

8 19 Further Notes and Views

2013-11-30T11:15:59Z

AnonMoos:

<noinclude>Back to [[8_19]].</noinclude>
[[8_19]] is the first non-homologically thin knot in the Rolfsen table. (That is, it's the first knot whose Khovanov homology has 'off-diagonal' elements.)

{|
|- valign=top
{{Knot View Template|
image = 8-19-knotscape.png|
text = Knotscape}}
|
{{Knot View Template|
image = 8crossing-symmetrical-nonalternating.png|
text = Symmetrical form}}
|}

File:8crossing-symmetrical-nonalternating.png

2013-11-30T11:12:18Z

AnonMoos: Self-made graphic, declared to be in public domain, modification of :Image:8crossing-symmetrical.png

Self-made graphic, declared to be in public domain, modification of [[:Image:8crossing-symmetrical.png]]

File:8-19-knotscape.png

2013-11-30T11:10:11Z

AnonMoos: Generated in Knotscape

Generated in Knotscape

8 16 Further Notes and Views

2013-06-25T07:01:59Z

AnonMoos: New page: <noinclude>Back to 8 16</noinclude> {| {{Knot View Template| image = 8-16 knot theory square.png | text = Square depiction.}} |}

<noinclude>Back to [[8 16]]</noinclude>
{|
{{Knot View Template|
image = 8-16 knot theory square.png |
text = Square depiction.}}
|}

File:8-16 knot theory square.png

2013-06-25T07:00:10Z

AnonMoos: Self-made graphic, declared to be in public domain, generated from the following PostScript source code: <pre>%! 1 setgray 18 setlinewidth/X{.5 add 72 mul exch .5 add 72 mul exch}def/l{X lineto}def 0 2 X moveto 5 2 l 5 3 l 2 3 l 2 0 l 4 0 l 4 4 l 1 4 l 1

Self-made graphic, declared to be in public domain, generated from the following PostScript source code:
<pre>%!
1 setgray 18 setlinewidth/X{.5 add 72 mul exch .5 add 72 mul
exch}def/l{X lineto}def 0 2 X moveto 5 2 l 5 3 l 2 3 l 2 0 l
4 0 l 4 4 l 1 4 l 1 1 l 3 1 l 3 5 l 0 5 l closepath gsave
0 setgray 36 setlinewidth stroke grestore stroke/r{72 mul exch
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2 2 4{/z exch def 0.5 2 2.5{dup 0.5 eq z 4 eq and{}{z X moveto
1 0 r stroke}ifelse}for}for
2 1.5 X moveto 0 1 r stroke 3 2.5 X moveto 0 1 r stroke
4 1.5 X moveto 0 1 r stroke 1 setgray 18 setlinewidth
1.48 1 X moveto 1.04 0 r stroke 3.48 3 X moveto 1.04 0 r stroke
2 2 4{/z exch def 0.48 2 2.48{dup 0.48 eq z 4 eq
and{}{z X moveto 1.04 0 r stroke}ifelse}for}for
2 1.48 X moveto 0 1.04 r stroke 3 2.48 X moveto 0 1.04 r stroke
4 1.48 X moveto 0 1.04 r stroke showpage
%EOF</pre>

L10a140 Quick Notes

2013-06-14T16:14:53Z

AnonMoos:

<noinclude>''Back to [[L10a140]]''<br></noinclude>
Brunnian link. Presumably the simplest [[Brunnian link]] other than the Borromean rings.[http://drorbn.net/AcademicPensieve/2010-08/nb/All%20Brunnians,%20Maybe.pdf] The second in an infinite series of Brunnian links -- if the blue and yellow loops in the illustration below have only one twist along each side, the result is the Borromean rings; if the blue and yellow loops have three twists along each side, the result is this L10a140 link; if the blue and yellow loops have five twists along each side, the result is a three-loop link with 14 overall crossings, etc.[http://www.mi.sanu.ac.rs/vismath/bor/bor4.htm]

L10a169 Further Notes and Views

2013-06-14T16:08:02Z

AnonMoos: New page: <noinclude>Back to L10a169</noinclude> {| {{Knot View Template| image = Yotsuwa Japanese mon.png | text = Japanese family symbol}} |}

<noinclude>Back to [[L10a169]]</noinclude>
{|
{{Knot View Template|
image = Yotsuwa Japanese mon.png |
text = Japanese family symbol}}
|}

File:Yotsuwa Japanese mon.png

2013-06-14T16:05:50Z

AnonMoos: Japanese ''mon'' (family symbol). Image declared to be public domain, see http://commons.wikimedia.org/wiki/File:Yotsuwa_inverted.png

Japanese ''mon'' (family symbol). Image declared to be public domain, see http://commons.wikimedia.org/wiki/File:Yotsuwa_inverted.png

L10a169 Quick Notes

2013-06-14T16:02:44Z

AnonMoos: New page: Compare L10n107.

Compare [[L10n107]].

L10n107 Quick Notes

2013-06-14T16:02:34Z

AnonMoos:

[[L10n107]] is the "Borromean chain mail" link - it contains two [[L6a4]] configurations without any [[L2a1]] configuration (i.e. no two loops are linked).

Compare [[L10a169]].

File:4intersecting-rounded-rectangles.png

2013-05-26T20:34:48Z

AnonMoos: new source

Self-made graphic, declared to be in public domain, generated from the following PostScript source code:
<pre>%!
306 396 translate .92 .96 1 setrgbcolor
21 setlinewidth/z{20000 sqrt 2 div}def
gsave 4{0 z 100 135 225 arc z 0 100 225 315 arc
z 2 mul 15 sub z 15 sub 100 315 45 arc
z 15 sub z 2 mul 15 sub 100 45 135 arc closepath
gsave 39 setlinewidth 0 setgray stroke grestore
stroke 90 rotate}repeat grestore 39 setlinewidth 0 setgray
0 z 100 213 225 arc 13 -13 rlineto stroke
21 setlinewidth .92 .96 1 setrgbcolor
0 z 100 211 225 arc 15 -15 rlineto stroke
gsave 2{39 setlinewidth 0 setgray z 0 100 269 302 arc stroke
21 setlinewidth .92 .96 1 setrgbcolor
z 0 100 267 304 arc stroke 90 rotate}repeat grestore
3{39 setlinewidth 0 setgray
z 15 sub z 2 mul 15 sub 100 107 135 arc -5 -5 rlineto stroke
21 setlinewidth .92 .96 1 setrgbcolor
z 15 sub z 2 mul 15 sub 100 105 135 arc -7 -7 rlineto stroke
90 rotate}repeat showpage
%EOF</pre>

Ornamental depictions of links with 12 crossings

2013-05-26T07:52:13Z

AnonMoos:

Some decorative depictions of links with 12 crossings:

<gallery>
Image:Celtic-knot-square-3loops.png|3-loop link in the form of a quasi-Celtic knot
Image:Two-interlaced-hexagrams.png|Two interlaced open hexagrams
Image:Brunnian-3-not-Borromean.gif|Three-component [[Brunnian link]]
Image:Brunnian-link-12crossings-nonBorromean-quasi-Arabesque.png|More ornate Brunnian link depiction
Image:Triquetra-double-interlaced.png|Two mutually-interlaced triquetras.
Image:Three-L4a1-config.png|Three-loop link with three [[L4a1]] configurations
Image:Three-L4a1.png|Two-loop link
Image:4squares-12crossings.png|Four linked squares
Image:4intersecting-rounded-rectangles.png|(variant)
Image:Four-loop-interlaced-circle.png|Four-fold looping interlaced with circle (two-loop link)
Image:Pleter50.png
Image:12-crossing-link Tallara Lousame Galicia Spain.jpg|Sculpture of preceding
Image:Cross of Uppland Runestone-.jpg|Ca. 1000 A.D. runestone
Image:Ccross-two-link.png|Quasi-Celtic two-loop link
Image:Six-rings.png|Six rings
Image:Six-link-closed-chain-Hisham-Umayyad-palace-Jericho.jpg|Decoration equivalent to a circle of six interlinked loops
Image:Trefoil-in-trefoil.png|Trefoil interlinked with smaller trefoil
Image:12-crossings-squares-16.png|3-loop link in square form
Image:Three-loop-link-12-crossings-alternate.png|(alternate version)
Image:12crossings-pseudo-Borromean.png|Three-loop link
Image:Triang-two-circ.png|Triangle and two circles
Image:12crossings-threesymmetric-link.png|Three loops symmetric
Image:12crossings-decorative-threefold-incircle-link.png|Three loops symmetric
Image:12crossing-twosymmetric-link.png|Two loops
Image:Borromean-plus-triangle.png|Four loops (Borromean + triangle)
Image:Link12-round.png|Two loops
Image:Two-trefoils-loop-12crossings.png|Three loops, two trefoils
Image:Two-trefoils-triangles-link.png|Two loops, two trefoils
Image:Two-trefoils-granny-12crossings.png|Two trefoils on one loop linked with another loop
Image:12crossings-pseudo-Celtic-link.png|One crossing of [[K13a1345]] eliminated, converting from knot to link...
Image:12crossing-link-inrectangle.png|Two interlinked trefoils inside rectangle
</gallery>

For ornamental depictions of knots with 12 crossings, see [[K12a477]], [[K12a503]], [[K12a541]], [[K12a561]], [[K12a975]], [[K12a991]], [[K12a1019]], [[K12a1210]], [[K12n242]], and (non-prime) [[:Image:Trefoil-of-trefoils.png]], [[:Image:Celtic-knot-insquare-green-transparentbg.png]], [[:Image:Three-figure8-knot triang1.png]], and [[:Image:Three-figure8-knot triang2.png]].

File:4intersecting-rounded-rectangles.png

2013-05-26T07:47:39Z

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