Knot Atlas - User contributions [en]

Main Page

2025-06-30T22:50:35Z

Drorbn:

=The Knot Atlas=
Welcome to the Knot Atlas! This site aims to be a complete user-editable knot atlas, in the [http://en.wikipedia.org/wiki/Wikimedia wiki] spirit of [http://en.wikipedia.org/wiki/Main_Page Wikipedia]. It is being developed primarily by [[User:Scott|Scott]] and [[User:Drorbn|Dror]], but any one can edit almost anything, anytime. Some advice can be found at [[how you can contribute]].

As with all evolving projects, the most important part of the Knot Atlas is the [[To Do]] list.

{| border=0 width=100%
|-
|width=120|[[Image:Rolfsen_240.png|120px|link=The Rolfsen Knot Table]]
|[[The Rolfsen Knot Table|The Rolfsen Table of knots with up to 10 crossings]].
[[The Hoste-Thistlethwaite Table of 11 Crossing Knots|The Hoste-Thistlethwaite Table of 11 Crossing Knots]].
|width=120|[[Image:Knots11_240.png|120px|link=The Hoste-Thistlethwaite Table of 11 Crossing Knots]]
|-
|[[Image:Links_240.png|120px|link=The Thistlethwaite Link Table]]
|[[The Thistlethwaite Link Table|The Thistlethwaite Link Table]].
[[36 Torus Knots|36 Torus Knots with up to 36 Crossings]].
|[[Image:TorusKnots_240.png|120px|link=36 Torus Knots]]
|-
|[[Image:KnotTheory_240.gif|120px|link=The Mathematica Package KnotTheory`]]
|[[The Mathematica Package KnotTheory`|The Mathematica Package KnotTheory`]].
[[The Take Home Database]]
|[[Image:TakeHomeDatabase.png|120px|link=The Take Home Database]]
|}

'''Extras''': Tube plots with [[Drawing with TubePlot|<code>TubePlot</code>]], [[WikiLink - The Mediawiki Interface]].

'''See also''': Jeremy Green's [http://www.math.toronto.edu/~drorbn/Students/GreenJ/ Table of Virtual Knots], [http://php.indiana.edu/~livingst/ Chuck Livingston's] amazing [https://knotinfo.org/ KnotInfo: Table of Knots], Hermann Gruber's [http://home.in.tum.de/~gruberh/ atlas of rational knots], [http://ipnweb.in2p3.fr/~lptms/membres/pzinn/ Paul Zinn-Justin's] [http://ipnweb.in2p3.fr/~lptms/membres/pzinn/virtlinks/ alternating virtual link database] and Slavik Jablan and Radmila Sazdanovic's [https://www.mi.sanu.ac.rs/vismath/linknot/ webMathematica LinKnot].

<center>

 

"God created the knots, all else in topology is the work of mortals."

[http://en.wikipedia.org/wiki/Kronecker Leopold Kronecker] (modified)

 

"Among Violet's many useful skills was a vast knowledge of different types of knots"

[http://en.wikipedia.org/wiki/Lemony_Snicket Lemony Snicket], A Series Of Unfortunate Events: The Bad Beginning
</center>

Setup

2024-05-05T09:45:24Z

Drorbn:

{{Manual TOC Sidebar}}

Start by downloading the file [https://drorbn.net/AcademicPensieve/Projects/KnotTheory/KnotTheory.zip <tt>KnotTheory.zip</tt>] (around 15MB), and unpack it. This will create a subdirectory KnotTheory/ in your current working directory. This done, no installation is required (though you may wish to check out [[#Further Data Files|Further Data Files]] and/or [[#Setting the Path|Setting the Path]] below). Start Mathematica and you're ready to go:



<tt>In[2]:=</tt><code> << KnotTheory`</code>

<tt>Loading KnotTheory` version of March 22, 2011, 21:10:4.67737.
Read more at http://katlas.org/wiki/KnotTheory.</tt>


Notice the little "prime" at the end of <tt>KnotTheory</tt> above. It is a backquote (find it on the upper left side of most keyboards) and not a quote, and it really has to be there for things to work.

Let us check that everything is working well:



{{InOut|
n = 3 |
in = <nowiki>Alexander[Knot[6, 2]][t]</nowiki> |
out= <nowiki> -2 3 2
-3 - t + - + 3 t - t
t</nowiki>}}




{{HelpLine|
n = 4 |
in = <nowiki>KnotTheoryVersion</nowiki> |
out= <nowiki>KnotTheoryVersion[] returns the date of the current version of the
package KnotTheory`. KnotTheoryVersion[k] returns the kth field in
KnotTheoryVersion[].</nowiki>}}




{{HelpLine|
n = 5 |
in = <nowiki>KnotTheoryVersionString</nowiki> |
out= <nowiki>KnotTheoryVersionString[] returns a string containing the date and
time of the current version of the package KnotTheory`. It is generated
from KnotTheoryVersion[].</nowiki>}}




{{HelpLine|
n = 6 |
in = <nowiki>KnotTheoryWelcomeMessage</nowiki> |
out= <nowiki>KnotTheoryWelcomeMessage[] returns a string containing the welcome message
printed when KnotTheory` is first loaded.</nowiki>}}


Thus on the day this manual page was last changed, we had:



{{InOut|
n = 7 |
in = <nowiki>{KnotTheoryVersion[], KnotTheoryVersionString[]}</nowiki> |
out= <nowiki>{{2011, 3, 22, 21, 10, 4.67737}, March 22, 2011, 21:10:4.67737}</nowiki>}}




{{InOut|
n = 8 |
in = <nowiki>KnotTheoryWelcomeMessage[]</nowiki> |
out= <nowiki>Loading KnotTheory` version of March 22, 2011, 21:10:4.67737.
Read more at http://katlas.org/wiki/KnotTheory.</nowiki>}}




{{HelpLine|
n = 9 |
in = <nowiki>KnotTheoryDirectory</nowiki> |
out= <nowiki>KnotTheoryDirectory[] returns the best guess KnotTheory` has for its
location on the host computer. It can be reset by the user.</nowiki>}}




{{InOut|
n = 10 |
in = <nowiki>KnotTheoryDirectory[]</nowiki> |
out= <nowiki>C:\Documents and Settings\pc\Documenti\Wolfram\KnotTheory</nowiki>}}


KnotTheoryDirectory may not work under some operating systems/environments. Please let [[User:Drorbn|Dror]] know if you encounter any difficulties.

===Notes===

====Precomputed Data====
<code>KnotTheory`</code> comes with a certain amount of precomputed data which is loaded "on demand" just when it is needed. When a precomputed data file is read by KnotTheory`, a notification message is displayed. To prevent these messages from appearing execute the command <code>Off[KnotTheory::loading]</code>.

====Further Data Files====
To access the Hoste-Thistlethwaite enumeration of knots with 12 to 16 crossings (see [[Naming and Enumeration]]), also download either the file [http://katlas.org/svn/KnotTheory/tags/stable/DTCodes4Knots12To16.tar.gz <tt>DTCodes4Knots12To16.tar.gz</tt>] or the file [http://katlas.org/svn/KnotTheory/tags/stable/DTCodes4Knots12To16.zip <tt>DTCodes4Knots12To16.zip</tt>] (about 9MB each), and unpack either one into the directory <tt>KnotTheory/</tt>.

====Setting the Path====
The directions above are written on the assumption that the package <tt>KnotTheory`</tt> (more precisely, the directory <tt>KnotTheory/</tt> containing the files that make this package), is somewhere on your Mathematica search path. Usually this will be the case if <tt>KnotTheory/</tt> is a subdirectory of your current working directory. If for some reason Mathematica cannot find <tt>KnotTheory`</tt>, you may tell it where to look in either of the following three ways. Assume <tt>KnotTheory/</tt> is a subdirectory of <tt>FullPathToKnotTheory</tt>:
# If you are using <tt>KnotTheory`</tt> rarely and you don't want to change system defaults, evaluate <tt>AppendTo[$Path,"FullPathToKnotTheory"]</tt> within Mathematica before attempting to load <tt>KnotTheory`</tt>.
# If you plan to use <tt>KnotTheory`</tt> often, you may want to move the directory <tt>KnotTheory/</tt> into one of the directories on your path. Evaluate <tt>$Path</tt> within Mathematica to see what those are.
# Alternatively, you may permanently add <tt>FullPathToKnotTheory</tt> to your <tt>$Path</tt>. To do that, find your Mathematica base directory by evaluating <tt>$UserBaseDirectory</tt> (on Dror's laptop, this comes out to be <tt>C:\Users\Dror\AppData\Roaming\Mathematica</tt>), and then add the line <tt>AppendTo[$Path,"FullPathToKnotTheory/"]</tt> to the file <tt>$BaseDirectory/Kernel/init.m</tt> and restart Mathematica.

Setup

2024-05-05T09:44:52Z

Drorbn:

{{Manual TOC Sidebar}}

Start by downloading the file [http://katlas.org/svn/KnotTheory/tags/stable/KnotTheory.zip <tt>KnotTheory.zip</tt>] (around 3MB each), and unpack either one. This will create a subdirectory KnotTheory/ in your current working directory. This done, no installation is required (though you may wish to check out [[#Further Data Files|Further Data Files]] and/or [[#Setting the Path|Setting the Path]] below). Start Mathematica and you're ready to go:



<tt>In[2]:=</tt><code> << KnotTheory`</code>

<tt>Loading KnotTheory` version of March 22, 2011, 21:10:4.67737.
Read more at http://katlas.org/wiki/KnotTheory.</tt>


Notice the little "prime" at the end of <tt>KnotTheory</tt> above. It is a backquote (find it on the upper left side of most keyboards) and not a quote, and it really has to be there for things to work.

Let us check that everything is working well:



{{InOut|
n = 3 |
in = <nowiki>Alexander[Knot[6, 2]][t]</nowiki> |
out= <nowiki> -2 3 2
-3 - t + - + 3 t - t
t</nowiki>}}




{{HelpLine|
n = 4 |
in = <nowiki>KnotTheoryVersion</nowiki> |
out= <nowiki>KnotTheoryVersion[] returns the date of the current version of the
package KnotTheory`. KnotTheoryVersion[k] returns the kth field in
KnotTheoryVersion[].</nowiki>}}




{{HelpLine|
n = 5 |
in = <nowiki>KnotTheoryVersionString</nowiki> |
out= <nowiki>KnotTheoryVersionString[] returns a string containing the date and
time of the current version of the package KnotTheory`. It is generated
from KnotTheoryVersion[].</nowiki>}}




{{HelpLine|
n = 6 |
in = <nowiki>KnotTheoryWelcomeMessage</nowiki> |
out= <nowiki>KnotTheoryWelcomeMessage[] returns a string containing the welcome message
printed when KnotTheory` is first loaded.</nowiki>}}


Thus on the day this manual page was last changed, we had:



{{InOut|
n = 7 |
in = <nowiki>{KnotTheoryVersion[], KnotTheoryVersionString[]}</nowiki> |
out= <nowiki>{{2011, 3, 22, 21, 10, 4.67737}, March 22, 2011, 21:10:4.67737}</nowiki>}}




{{InOut|
n = 8 |
in = <nowiki>KnotTheoryWelcomeMessage[]</nowiki> |
out= <nowiki>Loading KnotTheory` version of March 22, 2011, 21:10:4.67737.
Read more at http://katlas.org/wiki/KnotTheory.</nowiki>}}




{{HelpLine|
n = 9 |
in = <nowiki>KnotTheoryDirectory</nowiki> |
out= <nowiki>KnotTheoryDirectory[] returns the best guess KnotTheory` has for its
location on the host computer. It can be reset by the user.</nowiki>}}




{{InOut|
n = 10 |
in = <nowiki>KnotTheoryDirectory[]</nowiki> |
out= <nowiki>C:\Documents and Settings\pc\Documenti\Wolfram\KnotTheory</nowiki>}}


KnotTheoryDirectory may not work under some operating systems/environments. Please let [[User:Drorbn|Dror]] know if you encounter any difficulties.

===Notes===

====Precomputed Data====
<code>KnotTheory`</code> comes with a certain amount of precomputed data which is loaded "on demand" just when it is needed. When a precomputed data file is read by KnotTheory`, a notification message is displayed. To prevent these messages from appearing execute the command <code>Off[KnotTheory::loading]</code>.

====Further Data Files====
To access the Hoste-Thistlethwaite enumeration of knots with 12 to 16 crossings (see [[Naming and Enumeration]]), also download either the file [http://katlas.org/svn/KnotTheory/tags/stable/DTCodes4Knots12To16.tar.gz <tt>DTCodes4Knots12To16.tar.gz</tt>] or the file [http://katlas.org/svn/KnotTheory/tags/stable/DTCodes4Knots12To16.zip <tt>DTCodes4Knots12To16.zip</tt>] (about 9MB each), and unpack either one into the directory <tt>KnotTheory/</tt>.

====Setting the Path====
The directions above are written on the assumption that the package <tt>KnotTheory`</tt> (more precisely, the directory <tt>KnotTheory/</tt> containing the files that make this package), is somewhere on your Mathematica search path. Usually this will be the case if <tt>KnotTheory/</tt> is a subdirectory of your current working directory. If for some reason Mathematica cannot find <tt>KnotTheory`</tt>, you may tell it where to look in either of the following three ways. Assume <tt>KnotTheory/</tt> is a subdirectory of <tt>FullPathToKnotTheory</tt>:
# If you are using <tt>KnotTheory`</tt> rarely and you don't want to change system defaults, evaluate <tt>AppendTo[$Path,"FullPathToKnotTheory"]</tt> within Mathematica before attempting to load <tt>KnotTheory`</tt>.
# If you plan to use <tt>KnotTheory`</tt> often, you may want to move the directory <tt>KnotTheory/</tt> into one of the directories on your path. Evaluate <tt>$Path</tt> within Mathematica to see what those are.
# Alternatively, you may permanently add <tt>FullPathToKnotTheory</tt> to your <tt>$Path</tt>. To do that, find your Mathematica base directory by evaluating <tt>$UserBaseDirectory</tt> (on Dror's laptop, this comes out to be <tt>C:\Users\Dror\AppData\Roaming\Mathematica</tt>), and then add the line <tt>AppendTo[$Path,"FullPathToKnotTheory/"]</tt> to the file <tt>$BaseDirectory/Kernel/init.m</tt> and restart Mathematica.

Using the LinKnot package

2023-05-27T11:39:26Z

Drorbn:

{{Manual TOC Sidebar}}

The Mathematica package [https://www.mi.sanu.ac.rs/vismath/linknot/ ''LinKnot''] is a combination of two packages. The first, "Knot2000" was written by M. Ochiai and N. Imafuji. This was extended to the package ''LinKnot'' by S. Jablan and R. Sazdanovic. This package provides many useful things KnotTheory can't do natively; for example, it can interpret Conway notation, and, at least on Windows machines, provides nice graphics for knots, and a graphical interface for drawing knots by hand.

(''LinKnot'' also has a [http://math.ict.edu.yu/ webMathematica interface]!)

There are two ways ''LinKnot'' can be used along with <tt>KnotTheory`</tt>:
* As a "subcontractor": ''LinKnot'' is mostly hidden and wrapper functions are provided to access some of its functionality from within <tt>KnotTheory`</tt>.
* In parallel: everything from both packages is visible.

The advantage for the first approach is that the interface to ''LinKnot'' is then consistent with the conventions used by <tt>KnotTheory`</tt> and the Mathematica name space remains less cluttered. The advantage of the second approach is obvious - with it, all the functionality of ''LinKnot'' is available, and not just the parts for which wrapper functions are provided within <tt>KnotTheory`</tt>.

====Using ''LinKnot'' as a "subcontractor"====
To use ''LinKnot'' from within ''KnotTheory'', download the file <tt>LinKnot.zip</tt> from the [https://www.mi.sanu.ac.rs/vismath/linknot/ LinKnot site] or from [http://katlas.math.toronto.edu/svn/LinKnot/tags/stable/ our mirror]. Unzip and install the content of <tt>LinKnot.zip</tt> wherever you like, and within Mathematica issue a command like

{{Startup Note}}



{{In|
n = 2 |
in = <nowiki>AppendTo[$Path, "/path/to/LinKnots.m"];</nowiki>}}


After you've done this, everything should just work. (''LinKnot'' will only be loaded when necessary, and there might be a short delay when this happens.) If everything doesn't just work, please complain to [[User:Scott|Scott]].

In case of problems, you may find it helpful to explicitly set the path in which you've installed ''LinKnot'' (although ''KnotTheory'' tries to do this itself). Simply set <tt>LinKnotDirectory[]</tt>.



{{HelpLine|
n = 3 |
in = <nowiki>LinKnotDirectory</nowiki> |
out= <nowiki>LinKnotDirectory[] contains the path to the ''LinKnot'' package. It must be set correctly in order for all the (Windows only) MathLink components of ''LinKnot'' to be usable. It can be overriden by the user.</nowiki>}}


====Using ''LinKnot'' in parallel with <tt>KnotTheory`</tt>====
To use ''LinKnot'' in parallel with <tt>KnotTheory`</tt>, you should [https://www.mi.sanu.ac.rs/vismath/linknot/LinKnot.zip download LinKnot.zip] from the [https://www.mi.sanu.ac.rs/vismath/linknot/ LinKnot site] (a full ''LinKnot'' manual page ManualK2KC.nb is in the zip file, and the main file is the mathematica notebook <tt>K2KL.nb</tt>). After downloading the file LinKnot.zip:

# Extract LinKnot.zip anywhere (e.g., to the local disc "C:\"). It will automatically create a new folder <tt>LinKnot</tt>.
# Set the directory to <tt>LinKnot</tt>, add the path to <tt>KnotTheory`</tt> to the Mathematica <code>$Path</code>, and run:

<pre>
SetDirectory["/path/to/LinKnot/"];
<< LinKnots`
AppendTo[$Path, "Path to KnotTheory"];
<< KnotTheory`
</pre>

For example, if ''LinKnot'' is installed in "C:\LinKnot" and <tt>KnotTheory`</tt> is installed at "C:\KnotTheory", run:

<pre>
SetDirectory["C:\\LinKnot"];
<< LinKnots`
AppendTo[$Path, "C:\\"];
<< KnotTheory`
</pre>

Then you can work with the both programs <tt>KnotTheory`</tt> and ''LinKnot''.

If you need to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together <tt>KnotTheory`</tt> and ''LinKnot'', you can open the file <tt>K2KL.nb</tt> from the directory <tt>LinKnot</tt> and run the same command as before.

After you've done this, everything should just work. If everything doesn't just work, please complain to S. Jablan (jablans@yahoo.com).

====Further usage notes by [[User:Jablans|Jablans]]====

If you have any problem with running ''LinKnot'' (as a separate program) please do the following:

# in your ''Mathematica'' directory (e.g., C:\Program Files\Wolfram Research\Mathematica\5.0) make a new folder named LinKnot;
# extract the contents of the file LinKnot.zip into the directory LinKnot;
# open the "Mathematica" notebook K2KL.nb that you will find in the directory LinKnot;
# run the first line:

<pre>
SetDirectory["LinKnot"]
<< LinKnots.m
</pre>

Then you can fix the appropriate path and work with ''KnotTheory'' and ''LinKnot'' as well.

====See also====

See also [[Extending/Modifying KnotTheory`#Lessons learnt from integrating LinKnot`]] for technical details on how LinKnot` and KnotTheory` were integrated.

Main Page

2023-05-27T11:36:31Z

Drorbn: /* The Knot Atlas */

=The Knot Atlas=
Welcome to the Knot Atlas! This site aims to be a complete user-editable knot atlas, in the [http://en.wikipedia.org/wiki/Wikimedia wiki] spirit of [http://en.wikipedia.org/wiki/Main_Page Wikipedia]. It is being developed primarily by [[User:Scott|Scott]] and [[User:Drorbn|Dror]], but any one can edit almost anything, anytime. Some advice can be found at [[how you can contribute]].

As with all evolving projects, the most important part of the Knot Atlas is the [[To Do]] list.

{| border=0 width=100%
|-
|width=120|[[Image:Rolfsen_240.png|120px|link=The Rolfsen Knot Table]]
|[[The Rolfsen Knot Table|The Rolfsen Table of knots with up to 10 crossings]].
[[The Hoste-Thistlethwaite Table of 11 Crossing Knots|The Hoste-Thistlethwaite Table of 11 Crossing Knots]].
|width=120|[[Image:Knots11_240.png|120px|link=The Hoste-Thistlethwaite Table of 11 Crossing Knots]]
|-
|[[Image:Links_240.png|120px|link=The Thistlethwaite Link Table]]
|[[The Thistlethwaite Link Table|The Thistlethwaite Link Table]].
[[36 Torus Knots|36 Torus Knots with up to 36 Crossings]].
|[[Image:TorusKnots_240.png|120px|link=36 Torus Knots]]
|-
|[[Image:KnotTheory_240.gif|120px|link=The Mathematica Package KnotTheory`]]
|[[The Mathematica Package KnotTheory`|The Mathematica Package KnotTheory`]].
[[The Take Home Database]]
|[[Image:TakeHomeDatabase.png|120px|link=The Take Home Database]]
|}

'''Extras''': Tube plots with [[Drawing with TubePlot|<code>TubePlot</code>]], [[WikiLink - The Mediawiki Interface]].

'''See also''': Jeremy Green's [http://www.math.toronto.edu/~drorbn/Students/GreenJ/ Table of Virtual Knots], [http://php.indiana.edu/~livingst/ Chuck Livingston's] amazing [http://www.indiana.edu/~knotinfo/ Table of Knot Invariants], Hermann Gruber's [http://home.in.tum.de/~gruberh/ atlas of rational knots], [http://ipnweb.in2p3.fr/~lptms/membres/pzinn/ Paul Zinn-Justin's] [http://ipnweb.in2p3.fr/~lptms/membres/pzinn/virtlinks/ alternating virtual link database] and Slavik Jablan and Radmila Sazdanovic's [https://www.mi.sanu.ac.rs/vismath/linknot/ webMathematica LinKnot].

<center>

 

"God created the knots, all else in topology is the work of mortals."

[http://en.wikipedia.org/wiki/Kronecker Leopold Kronecker] (modified)

 

"Among Violet's many useful skills was a vast knowledge of different types of knots"

[http://en.wikipedia.org/wiki/Lemony_Snicket Lemony Snicket], A Series Of Unfortunate Events: The Bad Beginning
</center>

File:Képi Infanterie Sous-Lieutenant Armée Française.jpg

2016-10-18T22:01:56Z

Drorbn: Drorbn uploaded a new version of "File:Képi Infanterie Sous-Lieutenant Armée Française.jpg": A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

File:Képi Infanterie Sous-Lieutenant Armée Française.jpg

2016-10-18T22:01:54Z

Drorbn: Drorbn uploaded a new version of "File:Képi Infanterie Sous-Lieutenant Armée Française.jpg": A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

File:Képi Infanterie Sous-Lieutenant Armée Française.jpg

2016-10-18T21:59:50Z

Drorbn: Drorbn uploaded a new version of "File:Képi Infanterie Sous-Lieutenant Armée Française.jpg"

A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

File:Képi Infanterie Sous-Lieutenant Armée Française.jpg

2016-10-18T21:59:14Z

Drorbn: Drorbn uploaded a new version of "File:Képi Infanterie Sous-Lieutenant Armée Française.jpg": Reverted to version as of 11:10, 26 September 2016

A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

File:Képi Infanterie Sous-Lieutenant Armée Française.jpg

2016-10-18T21:58:41Z

Drorbn: Drorbn uploaded a new version of "File:Képi Infanterie Sous-Lieutenant Armée Française.jpg"

A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

8 18 Further Notes and Views

2016-09-26T11:13:37Z

Drorbn:

<noinclude>Back to [[8_18]].</noinclude>
 
{|
|- 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.
}}
|}

File:Képi Infanterie Sous-Lieutenant Armée Française.jpg

2016-09-26T11:10:55Z

Drorbn: A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

A "Hungarian Knot", decorating French Military uniforms. By Jean-Marie Nicolas.

User talk:Drorbn

2016-07-13T16:35:39Z

Drorbn: /* You have a spammer */

==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)
 

==(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)

10 161 Further Notes and Views

2016-04-30T13:56:20Z

Drorbn:

<noinclude>Back to [[10 161]]</noinclude> 

'''Warning.''' In 1973 K. Perko noticed that the knots that were later labeled 10161 and 10162 in Rolfsen's tables (which were published in 1976 and were based on earlier tables by Little (1900) and Conway (1970)) are in fact the same. In our table we removed Rolfsen's 10162 and renumbered the subsequent knots, so that our 10 crossings total is 165, one less than Rolfsen's 166. Read more: [http://www.math.cuhk.edu.hk/publect/lecture4/perko.html] [http://www.maths.warwick.ac.uk/~bjs/Perko-page.html] [http://www.math.uiuc.edu/~jms/Videos/ke/images.html]
[http://knotplot.com/perko/].

User talk:AnonMoos

2016-04-18T17:15:19Z

Drorbn: /* Editing large knot pages. */

See [[The_Multivariable_Alexander_Polynomial#Detecting_a_Link_Using_the_Multivariable_Alexander_Polynomial]]. --[[User:Drorbn|Drorbn]] 16:16, 3 May 2007 (EDT)

See the bottom of [[Heegaard Floer Knot Homology]]. --[[User:Drorbn|Drorbn]] 12:25, 20 December 2007 (EST)

== Editing large knot pages. ==

Dear AnonMoos,

I think it is a bad idea to edit large knot pages (e.g. [[K12a1019]]) directly. The knot pages themselves were created by robots, and may be overwritten the next time the robot runs. The right thing to do is to add a "quick notes" handle of some kind to [[Template:Large Knot Page]], in a manner similar to the small knot pages, and then edit the relevant "quick notes" link.

Best,

[[User:Drorbn|Drorbn]] 06:07, 25 February 2010 (EST)

== Thumbnailing ==

Dear AnonMoos,

Sorry for taking long to respond to your comment from March 21. Scott and I took a look and found no problem. Can you be more specific?

Best,

--[[User:Drorbn|Drorbn]] ([[User talk:Drorbn|talk]]) 13:15, 18 April 2016 (EDT)

9 23 Further Notes and Views

2016-02-01T16:57:03Z

Drorbn:

<noinclude>Back to [[9 23]].</noinclude>
 
{|
|- valign=top
{{Knot View Template|
image = 9crossings-2symmetrical.png |
text = Symmetrical decorative knot}}
|{{Knot View Template|
image = 9crossing-knot symmetrical grid.png |
text = With crossings on 3x3 grid}}
|{{Knot View Template|
image = Cjdfjjhd.png |
text = Depiction with two axes of symmetry}}
|{{Knot View Template|
image = mongolian632.2.gif |
text = Mongolian ornament (two crossings are unnecessary)}}
|- valign=top
{{Knot View Template|
image = mongolian9.23.gif |
text = Mongolian ornament, sum of two 9.23}}
|{{Knot View Template|
image = logo-icmc.jpg |
text = Logo of the [http://www.icmc.usp.br/ ICMC-USP], Brazil}}
|}

9 23 Further Notes and Views

2016-02-01T16:03:13Z

Drorbn:

<noinclude>Back to [[9 23]].</noinclude>
 
{|
|- valign=top
{{Knot View Template|
image = 9crossings-2symmetrical.png |
text = Symmetrical decorative knot}}
|
{{Knot View Template|
image = 9crossing-knot symmetrical grid.png |
text = With crossings on 3x3 grid}}
{{Knot View Template|
image = Cjdfjjhd.png |
text = Depiction with two axes of symmetry}}
{{Knot View Template|
image = mongolian632.2.gif |
text = Mongolian ornament (two crossings are unnecessary)}}
{{Knot View Template|
image = mongolian9.23.gif |
text = Mongolian ornament, sum of two 9.23}}
{{Knot View Template|
image = logo-icmc.jpg |
text = Logo of the [http://www.icmc.usp.br/ ICMC-USP], Brazil}}
|}

9 23 Further Notes and Views

2016-02-01T16:02:58Z

Drorbn:

<noinclude>Back to [[9 23]].</noinclude>
 
{|
|- valign=top
{{Knot View Template|
image = 9crossings-2symmetrical.png |
text = Symmetrical decorative knot}}
|
{{Knot View Template|
image = 9crossing-knot symmetrical grid.png |
text = With crossings on 3x3 grid}}
{{Knot View Template|
image = Cjdfjjhd.png |
text = Depiction with two axes of symmetry}}
{{Knot View Template|
image = mongolian632.2.gif |
text = Mongolian ornament (two crossings are unnecessary)}}
{{Knot View Template|
image = mongolian9.23.gif |
text = Mongolian ornament, sum of two 9.23}}
{{Knot View Template|
image = logo-icmc.jpg |
text = Logo of the Logo of the [http://www.icmc.usp.br/ ICMC-USP], Brazil}}
|}

File:Logo-icmc.jpg

2016-02-01T16:02:10Z

Drorbn:

Logo of the [http://www.icmc.usp.br/ ICMC-USP] - Instituto de Ciências Matemáticas e de Computação of the University of São Paulo, Brazil (Institute of Mathemathical and Computational Sciences).

File:Logo-icmc.jpg

2016-02-01T16:01:35Z

Drorbn:

Logo of the [http://www.icmc.usp.br/ ICMC-USP] - Instituto de Ciências Matemáticas e de Computação of the University of São Paulo, Brazil, acronym (Institute of Mathemathical and Computational Sciences).

File:Logo-icmc.jpg

2016-02-01T16:00:31Z

Drorbn: Logo of the [http://www.icmc.usp.br/ICMC-USP] - Instituto de Ciências Matemáticas e de Computação of the University of São Paulo, Brazil, acronym (Institute of Mathemathical and Computational Sciences).

Logo of the [http://www.icmc.usp.br/ICMC-USP] - Instituto de Ciências Matemáticas e de Computação of the University of São Paulo, Brazil, acronym (Institute of Mathemathical and Computational Sciences).

User talk:Robert FERREOL

2015-09-01T17:35:35Z

Drorbn: /* Sep 1, 2015 */ new section

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.

L8a14 Further Notes and Views

2015-09-01T17:30:48Z

Drorbn: Reverted edits by Robert FERREOL (talk) to last revision by 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 Poertugal}}
|
{{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}}
|}

Template:Links Khovanov Homology

2015-07-03T05:17:49Z

Drorbn: Drorbn moved page Template:Links Khovanov Homology to Template:Link Khovanov Homology

#REDIRECT [[Template:Link Khovanov Homology]]

Template:Link Khovanov Homology

2015-07-03T05:17:49Z

Drorbn: Drorbn moved page Template:Links Khovanov Homology to Template:Link Khovanov Homology

<div style="border: solid pink 1px">
===[[Khovanov Homology]]===

{| width=98%
|- valign=top
|width=30%|The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>).
|  
|{{Data:{{PAGENAME}}/KhovanovTable}}
|}

{| width=98%
|- valign=top
|width=30%|[[Khovanov Homology|Integral Khovanov Homology]]
([[Data:{{PAGENAME}}/Integral Khovanov Homology|db]], [[Integral Khovanov Homology data source|data source]])
|  
|{{Data:{{PAGENAME}}/Integral Khovanov Homology}}
|}
</div>

Template:Link Page

2015-07-03T05:15:47Z

Drorbn:

{{Knot Navigation Links|ext=gif}}
{{TOCright}}

{|
|-
| valign="top" rowspan=2 align=center| [[Image:{{PAGENAME}}.gif]] ([[Further_Knot_Theory_Software#Knotscape|Knotscape]] image)
|See the full [[The Thistlethwaite Link Table|Thistlethwaite Link Table]] (up to 11 crossings).
Visit [http://knotilus.math.uwo.ca/draw.php?knot={{urlencode:{{Data:{{PAGENAME}}/Gauss Code}}}} {{PAGENAME}}] at [http://knotilus.math.uwo.ca/ Knotilus]!
|-
| valign="top" | {{floating edit link|{{PAGENAME}} Quick Notes}}
{{#ifexist:{{PAGENAME}} Quick Notes|{{:{{PAGENAME}} Quick Notes}}|}}
|}

{{floating edit link|{{PAGENAME}} Further Notes and Views}}
{{#ifexist:{{PAGENAME}} Further Notes and Views|{{:{{PAGENAME}} Further Notes and Views}}|}}

<div style="border: solid pink 1px">
===Link Presentations===

{{edit link|Notes on {{PAGENAME}}'s Link Presentations}}
{{#ifexist:Notes on {{PAGENAME}}'s Link Presentations|{{:Notes on {{PAGENAME}}'s Link Presentations}}|}}
{|
|- valign=top
|'''[[Planar Diagrams|Planar diagram presentation]]'''
|style="padding-left: 1em; word-wrap: break-word" | {{Data:{{PAGENAME}}/PD Presentation}}
|- valign=top
|'''[[Gauss Codes|Gauss code]]'''
|style="padding-left: 1em; word-wrap: break-word" | {{Data:{{PAGENAME}}/Gauss Code}}
|}
{|
|- valign=top
|'''[[Braid Representatives|A Braid Representative]]'''
|style="padding-left: 1em;" | {{{braid_table}}}
|- valign=top
|'''[[MorseLink Presentations|A Morse Link Presentation]]'''
|style="padding-left: 1em;" | [[Image:{{PAGENAME}}_ML.gif]]
|}
</div>
{{Link Polynomial Invariants}}
{{Link Khovanov Homology|table={{{khovanov_table}}}}}
{{Computer Talk Header}}
<div style="border: solid pink 1px">

=== Modifying This Page ===
{| width=100%
|- valign=bottom
|align=left|'''Read me first:''' [[Modifying Knot Pages]]

See/edit the [[Template:Link Page|Link Page]] master template (intermediate).

See/edit the [[Link_Splice_Base]] (expert).

Back to the [[#top|top]].
|align=right|{{Knot Navigation Links|ext=gif}}
|}
</div>

[[Category:Knot Page]]

Template:Link Khovanov Homology

2015-07-03T05:15:23Z

Drorbn: Created page with "<div style="border: solid pink 1px"> ===Khovanov Homology=== {| width=98% |- valign=top |width=30%|The coefficients of the monomials <math>t^rq^j</math> are shown, along ..."

Sandbox

2014-04-27T21:40:13Z

Drorbn:

Zubi

Sandbox

2014-04-27T13:55:40Z

Drorbn:

Test suitcases.

Sandbox

2014-04-26T22:09:30Z

Drorbn: Created page with "Test"

Test

CableComponent.m

2013-10-20T19:09:11Z

Drorbn:

(*

The program <code>CableComponent</code>, documented in the page on [[Cabling]]. It is not part of the package <code>KnotTheory`</code> but is designed to work with it. Use <code><nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"]</nowiki></code> to download into a mathematica session, or copy-paste the text below ignoring the <code>*)</code> on the first line and the <code>(*</code> on the last.
<pre>
*)
CableComponent[BR[n_Integer, js_List], K_] :=
Module[{BC, C0, C1, C2, CC1, CS1, CS2, L, S, a, e, h, i, i1, i2, j,
j1, j2, jss, k, k1, kjs, out, out0, out1, p, p1, pos, q, r, s, ss,
t, t0, t1, t2, tj, v, w, writhe},
L = PD[K];
kjs = BR[L][[2]];
For[i1 = 1; writhe = 0, i1 <= Length[kjs], i1++,
writhe = writhe + Sign[kjs[[i1]]]];
For[i2 = 1; jss[0] = js, i2 <= n Abs[writhe], i2++,
jss[i2] =
Flatten[{jss[i2 - 1], Table[-Sign[writhe] e, {e, n - 1}]}]];
k1 = Length[jss[n Abs[writhe]]];
For[i = 1, i <= n, i++, s[i] = a[i] = i];
For[
j = 1, j <= k1, j++,
p = Select[Range[n], Abs[jss[n Abs[writhe]][[j]]] == a[#] &][[
1]];
q = Select[Range[n], a[#] == a[p] + 1 &][[1]];
If[jss[n Abs[writhe]][[j]] > 0,
K[j] = X[s[q], n + 2 j, n + 2 j - 1, s[p]],
K[j] = X[s[p], s[q], n + 2 j, n + 2 j - 1]];
s[p] = n + 2 j;
s[q] = n + 2 j - 1;
a[p]++;
a[q]--
];
BC = Table[K[d], {d, k1}];
If[Jones[L][q] === 1,
For[j1 = 1, j1 <= Length[BC], j1++,
For[i = 1, i <= n, i++, BC[[j1]] = BC[[j1]] /. s[i] :> a[i]
]];
If[BC == {}, BC = {Loop[1]}];
out1 = PD @@ BC,
For[j2 = 1, j2 <= Length[BC], j2++,
For[tj = 1, tj <= n, tj++, BC[[j2]] = BC[[j2]] /. tj :> 1[tj]]
];
p1 = Select[Range[n], # != s[#] &];
S = Select[L, MemberQ[#, 1] && MemberQ[#, 2] & ];
pos = Position[S, 1][[1, 2]];
r = Select[Table[i, {i, Length[L]}], L[[#]] == Flatten @@ S &][[
1]];
k = 0;
out0 = L /. X[a_, b_, c_, d_] :> (
++k;
Table[
X[h[i, j - 1, k], v[i, j, k], h[i, j, k], v[i - 1, j, k]],
{i, n}, {j, n}
] /. {h[i_, 0, _] :> a[i], h[i_, n, _] :> c[i]} /. If[
d - b == 1 || b - d > 1,
{v[0, j_, _] :> d[j],
v[n, j_, _] :> b[j]}, {v[0, j_, _] :> d[n + 1 - j],
v[n, j_, _] :> b[n + 1 - j]}
]
);
w = Flatten@out0[[r]];
out = PD @@ Flatten[Join @@ out0];
ss = Table[a[i], {i, n}][[p1]];
CC1 = List @@ out;
For[t0 = 1, t0 <= Length[ss], t0++,
C0[t0] = Select[w, MemberQ[#, 1[ss[[t0]]]] &];
C1[t0] =
Select[C0[t0],
Mod[Position[#, 1[ss[[t0]]]][[1, 1]], 2] == Mod[pos, 2] &];
C2[t0] =
C1[t0] /. 1[ss[[t0]]] :>
s[Select[Range[n], a[#] == ss[[t0]] &][[1]]]];

CS1 = Flatten[Table[C1[t1], {t1, Length[ss]}]];
CS2 = Flatten[Table[C2[t2], {t2, Length[ss]}]];
For[i = 1, i <= Length[CS1], i++,
CC1 = DeleteCases[CC1, CS1[[i]]]];
out1 = Union[BC, CC1, CS2];
PD @@ out1;
k = 0;
out1 =
PD @@ ( out1 /. ((# -> ++k) & /@ (List @@ Union @@ out1)))]];
(* </pre> *)

SubLink.m

2013-10-20T19:08:53Z

Drorbn:

(*

The program <code>SubLink</code>, documented in the page on [[Prime Links with a Non-Prime Component]]. It is not part of the package <code>KnotTheory`</code> but is designed to work with it. Use <code><nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"]</nowiki></code> to download into a mathematica session, or copy-paste the text below ignoring the <code>*)</code> on the first line and the <code>(*</code> on the last.
<pre>
*)
SubLink::usage = "SubLink[pd, js] returns the
sublink of pd made out of the components of pd in the list js.";
SubLink[pd_PD, js_List] := Module[
{k, t0, t, t1, t2, s0, s1},
s0 = Skeleton[pd];
(* t0 contains the list of edges that should appear in the sublink *)
t0 = Flatten[List @@@ s0[[js]]];
(* t is pd with all edges not in t0 removed;
this means that some crossings will now involve just 0 or 2 edges. *)
t = pd /. x_X :> Select[x, MemberQ[t0, #] &];
(* Remove all "empty" crossings from t: *)
t = DeleteCases[t, X[] | Loop[]];
(* Remove all "valency 2" crossings from t,while also removing not-
longer-necessary edge labels: *)
k = 1;
While[
k <= Length[t],
If[Length[t[[k]]] == 2,
t = Delete[t, k] /. (Rule @@ t[[k]]),
(* else *) ++k
];
];
(* We have to manually "re-add" all skeleton components that "disappeared": *)
s1 = Union[Flatten[List @@ List @@@ t]];
Do[
If[
MemberQ[js, k] && (And @@ (FreeQ[s1, #] & /@ s0[[k]])),
AppendTo[t, Loop[s0[[k, 1]]]];
AppendTo[s1, s0[[k, 1]]]
],
{k, Length[s0]}
];
(* t1 will have all edge-labels still appearing in t;
it is used to relabel t so that the edge labels will be consecutive *)
t1 = Sort[s1];
t2 = Thread[(t1) -> Range[Length[t1]]];
t /. t2
];
SubLink[pd_PD, j_] := SubLink[pd, {j}];
SubLink[L_, js_] := SubLink[PD[L], js];
(* </pre> *)

Burau's Theorem

2013-10-20T19:07:54Z

Drorbn:

{{Manual TOC Sidebar}}

An interesting property of the Alexander polynomial related to cables is Burau's theorem which says the following: If we take the <math>n</math>-th cable of a knot from the Knot Atlas and insert in it a braid with <math>1/n</math> of a full twist, the Alexander polynomial of the result with respect to <math>t</math> is the same as the Alexander polynomial of the original knot with respect to <math>t^n</math>. This can only be seen if we take into consideration the writhe <math>w</math> of the knot and add the appropriate number of twists (<math>nw(1/n)=w</math> full twists) in the direction opposite to the sign of the writhe. We can test the theorem using the program [[Cabling|CableComponent]] and performing the above operation on knot [[K11n152]], for example.

{{Startup Note}}



{{In|
n = 2 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];</nowiki>}}




{{Graphics|
n = 3 |
in = <nowiki>(cc = CableComponent[BR[6, {1, 2, 3, 4, 5}],
K = Mirror[Knot[11, NonAlternating, 52]]]) // DrawMorseLink
</nowiki> |
img= Buraus_Theorem_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}

It is not immediately clear from the diagram that this is the expected result but we can see that the Alexander polynomial relation holds:


{{InOut|
n = 4 |
in = <nowiki>Alexander[#][t] & /@ {cc, K}</nowiki> |
out= <nowiki> -18 6 14 6 12 18
{-17 + t - --- + -- + 14 t - 6 t + t ,
12 6
t t

-3 6 14 2 3
-17 + t - -- + -- + 14 t - 6 t + t }
2 t
t</nowiki>}}

Cabling

2013-10-20T19:06:51Z

Drorbn:

{{Manual TOC Sidebar}}

<code>CableComponent[BR[n,js],K]</code>, whose code is available [[CableComponent.m|here]], returns the <math>n</math>-th cable of the knot <math>K</math> with the braid on <math>n</math> strands with crossings <code>js = {j1, j2, ...}</code> inserted in it. It also performs the necessary number of <math>1/n</math>-twists on the components of the cable to compensate for a non-zero writhe number of the original knot. Cabling knot [[3_1]], for instance, and inserting the braid <code>BR[3,{1,2}]</code>, we get:

{{Startup Note}}


{{In|
n = 2 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];</nowiki>}}



{{Graphics|
n = 3 |
in = <nowiki>CableComponent[BR[3, {1, 2}], Knot[3, 1]] // DrawMorseLink</nowiki> |
img= Cabling_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}


For some special cases, we can check our result using [[Burau's Theorem]].

Cabling

2013-10-20T19:06:02Z

Drorbn:

{{Manual TOC Sidebar}}

<code>CableComponent[BR[n,js],K]</code>, whose code is available [[CableComponent.m|here]], returns the <math>n</math>-th cable of the knot <math>K</math> with the braid on <math>n</math> strands with crossings <code>js = {j1, j2, ...}</code> inserted in it. It also performs the necessary number of <math>1/n</math>-twists on the components of the cable to compensate for a non-zero writhe number of the original knot. Cabling knot [[3_1]], for instance, and inserting the braid <code>BR[3,{1,2}]</code>, we get:

{{Startup Note}}


{{In|
n = 2 |
in = <nowiki>Import["http://katlas.org/w/index.php/title=CableComponent.m&action=raw"];</nowiki>}}



{{Graphics|
n = 3 |
in = <nowiki>CableComponent[BR[3, {1, 2}], Knot[3, 1]] // DrawMorseLink</nowiki> |
img= Cabling_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}


For some special cases, we can check our result using [[Burau's Theorem]].

Identifying Knots within a List

2013-10-20T19:04:46Z

Drorbn:

{{Manual TOC Sidebar}}

<code>IdentifyWithin[L,H]</code>, whose code is available [[IdentifyWithin.m|here]], returns those elements from the list of knots <math>H</math>, whose invariant matches that of the knot <math>L</math>. It can also recognize mirrors and connected sums of the knots in the list. Its options include turning off (on) the search for connected sums with <code>ConnectedSum->False (True)</code> and choosing the invariants to be used in identification by selecting, for example, <code>Invariants->{Jones[#][q]&, HOMFLYPT[#][a,z]&}</code>.
<code>IdentifyWithin</code> can be used together with [[Prime Links with a Non-Prime Component|<code>SubLink</code>]] to determine the components of a link. For the second component of link [[L11n150]], for instance, we get:

{{Startup Note}}



{{In|
n = 2 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=IdentifyWithin.m&action=raw"];</nowiki>}}




{{In|
n = 3 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];</nowiki>}}




{{InOut|
n = 4 |
in = <nowiki>IdentifyWithin[SubLink[Link["L11n150"], 2], AllKnots[]]</nowiki> |
out= <nowiki>{Knot[5, 2]}</nowiki>}}


{{Knot Image Pair|L11n150|gif|5_2|gif}}

Unfortunately, the program does not provide absolute identification when all the used invariants cannot distinguish between two or more different knots. In that case, a list of possible candidates for <math>L</math> is returned.

"Rubberband" Brunnian Links

2013-10-20T19:03:30Z

Drorbn:

{{Manual TOC Sidebar}}

A "Rubberband" [[Brunnian link]] is obtained by connecting unknots in a closed chain as illustrated in the diagram of the 10-component link, where the last knot gets connected to the first one.

{|
|-
|[[Image:RubberBand_Link_10.PNG|220px|The Rubberband link with 10 components]]
|[[Image:Brunnian_Link_Example.PNG|300px]]
|}

If we number the strands in one section of the link as shown and proceed with numbering each following section in the same manner, we can get its PD form. The PD of any "Rubberband" link can be generated in this way by varying the desired number of components:

{{Startup Note}}



{{In|
n = 1 |
in = <nowiki>K0 =
PD[X[1, 10, 5, 12], X[2, 12, 6, 14], X[5, 11, 8, 13],
X[6, 13, 9, 15], X[10, 0, 16, 4], X[11, 4, 17, 8], X[14, 7, 19, 3],
X[15, 9, 18, 7]];</nowiki>}}




{{In|
n = 2 |
in = <nowiki>RubberBandBrunnian[n_] :=
Join @@ Table[K0 /. j_Integer :> j + 16 k, {k, 0, n - 1}] /. {16
n -> 0, 16 n + 1 -> 1, 16 n + 2 -> 2, 16 n + 3 -> 3}</nowiki>}}


For instance, let us draw the links with three, four, and five components and compute their Jones polynomials:



{{Graphics|
n = 4 |
in = <nowiki>DrawMorseLink[RBB3=RubberBandBrunnian[3]]</nowiki> |
img= Rubberband_Brunnian_Links_Out_3.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>Jones[RBB3][q]</nowiki> |
out= <nowiki> 2 3 4 5 7 8 9 10
-q + 5 q - 11 q + 14 q - 10 q + 11 q - 18 q + 24 q - 18 q +

11 13 14 15 16 17
11 q - 10 q + 14 q - 11 q + 5 q - q</nowiki>}}




{{Graphics|
n = 7 |
in = <nowiki>DrawMorseLink[RBB4=RubberBandBrunnian[4]]</nowiki> |
img= Rubberband_Brunnian_Links_Out_6.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 8 |
in = <nowiki>Jones[RBB4][q]</nowiki> |
out= <nowiki> 3/2 5/2 7/2 9/2 11/2 13/2 15/2
-q + 7 q - 24 q + 49 q - 56 q + 18 q + 51 q -

17/2 19/2 21/2 23/2 25/2 27/2
111 q + 131 q - 100 q + 32 q + 32 q - 100 q +

29/2 31/2 33/2 35/2 37/2 39/2
131 q - 111 q + 51 q + 18 q - 56 q + 49 q -

41/2 43/2 45/2
24 q + 7 q - q</nowiki>}}




{{Graphics|
n = 10 |
in = <nowiki>DrawMorseLink[RBB5=RubberBandBrunnian[5]]</nowiki> |
img= Rubberband_Brunnian_Links_Out_9.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 11 |
in = <nowiki>Jones[RBB5][q]</nowiki> |
out= <nowiki> 2 3 4 5 6 7 8 9
-q + 9 q - 40 q + 110 q - 189 q + 167 q + 57 q - 414 q +

10 11 12 13 14 15 16
660 q - 581 q + 189 q + 305 q - 672 q + 816 q - 672 q +

17 18 19 20 21 22 23
305 q + 189 q - 581 q + 660 q - 414 q + 57 q + 167 q -

24 25 26 27 28
189 q + 110 q - 40 q + 9 q - q</nowiki>}}


We can also check that when one component is removed the remaining link is trivial:



{{In|
n = 12 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];</nowiki>}}




{{In|
n = 13 |
in = <nowiki>S = SubLink[RubberBandBrunnian[5], {1, 2, 3, 4}];</nowiki>}}




{{InOut|
n = 14 |
in = <nowiki>J=Factor[Jones[S][q]]</nowiki> |
out= <nowiki> 6 3
-(q (1 + q) )</nowiki>}}


Similarly, in the case of Brunnian braids, removing one strand gives us a trivial braid. We can verify that using the following two programs. The first one constructs a Brunnian braid while the second one removes a selected strand:



{{In|
n = 15 |
in = <nowiki>BR /: Inverse[BR[n_, l_List]] := BR[n, -Reverse[l]];
BR /: BR[n1_, l1_] ** BR[n2_, l2_] := BR[Max[n1, n2], Join[l1, l2]];
BrunnianBraid[2] = BR[2, {1, 1}];
BrunnianBraid[n_] /; n > 2 := Module[
{b0},
b0 = BrunnianBraid[n - 1];
((b0 ** BR[n, {n - 1, n - 1}]) ** Inverse[b0]) **
BR[n, {1 - n, 1 - n}]
]</nowiki>}}




{{In|
n = 16 |
in = <nowiki>DeleteStrand[k_, BR[n_, l_List]] := BR[n - 1, DeleteStrand[k, l]];
DeleteStrand[k_, {}] = {};
DeleteStrand[k_, {j1_, js___}] := Which[
k < Abs[j1], {j1 - Sign[j1]}~Join~DeleteStrand[k, {js}],
k == Abs[j1], DeleteStrand[k + 1, {js}],
k == Abs[j1] + 1, DeleteStrand[k - 1, {js}],
k > Abs[j1] + 1, {j1}~Join~DeleteStrand[k, {js}]
]</nowiki>}}


Testing for the Brunnian braid with four strands, we get:



{{Graphics|
n = 18 |
in = <nowiki>(b = BrunnianBraid[4]) // BraidPlot </nowiki> |
img= Rubberband_Brunnian_Links_Out_17.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 19 |
in = <nowiki>Jones[b][q]</nowiki> |
out= <nowiki> -(11/2) 4 6 5 5 1 3/2
-q + ---- - ---- + ---- - ---- - ------- - Sqrt[q] - 5 q +
9/2 7/2 5/2 3/2 Sqrt[q]
q q q q

5/2 7/2 9/2 11/2
5 q - 6 q + 4 q - q</nowiki>}}




{{Graphics|
n = 21 |
in = <nowiki>(bb = DeleteStrand[4, b]) // BraidPlot</nowiki> |
img= Rubberband_Brunnian_Links_Out_20.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 22 |
in = <nowiki>Jones[#][q] & /@ {bb, BR[3, {}]}</nowiki> |
out= <nowiki> 1 1
{2 + - + q, 2 + - + q}
q q</nowiki>}}

Prime Links with a Non-Prime Component

2013-10-20T19:00:25Z

Drorbn:

{{Manual TOC Sidebar}}
Let us find all (prime!) links in the Knot Atlas that have a non-prime component. Since the links listed in the Knot Atlas have at most 11 crossings, such a component may only be the sum of exactly two knots chosen among the trefoil, the figure eight knot, and their mirror images. The figure eight knot's mirror image is itself so we have five possibilities. Computing the Jones polynomial of each, we get:

{{Startup Note}}



{{In|
n = 2 |
in = <nowiki>K31 = Knot[3, 1]; K41 = Knot[4, 1];</nowiki>}}




{{InOut|
n = 3 |
in = <nowiki>CompositeJones =
Jones[#][q] & /@ {ConnectedSum[K31, K31],
ConnectedSum[K31, Mirror[K31]],
ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],
ConnectedSum[Mirror[K31], K41]}</nowiki> |
out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3
{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,
7 5 4 q
q q q

2 4 5 6 7 8
q + 2 q - 2 q + q - 2 q + q ,

-6 2 2 3 3 2
-1 - q + -- - -- + -- - -- + - + q,
5 4 3 2 q
q q q q

1 2 3 4 5 6
-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q }
q</nowiki>}}


Now, we can use the program [[SubLink.m|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:



{{In|
n = 4 |
in = <nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];</nowiki>}}


Using <code>SubLink</code> and the Jones polynomials of the five composite knots mentioned above, we can find all links that have one of these as a component:



{{In|
n = 5 |
in = <nowiki>NonPrimeComponentQ[L_] :=
Or @@ (MemberQ[CompositeJones, Jones[SubLink[L, #]][q]] & /@
Range[Length[Skeleton[L]]])</nowiki>}}




{{InOut|
n = 6 |
in = <nowiki>Exceptions= Select[AllLinks[], NonPrimeComponentQ]</nowiki> |
out= <nowiki>{Link[10, Alternating, 38], Link[10, Alternating, 39],

Link[10, Alternating, 46], Link[10, NonAlternating, 35],

Link[10, NonAlternating, 36], Link[10, NonAlternating, 37],

Link[10, NonAlternating, 38], Link[10, NonAlternating, 39],

Link[11, Alternating, 91], Link[11, Alternating, 92],

Link[11, Alternating, 93], Link[11, Alternating, 95],

Link[11, Alternating, 121], Link[11, Alternating, 128],

Link[11, Alternating, 130], Link[11, NonAlternating, 110],

Link[11, NonAlternating, 111], Link[11, NonAlternating, 112],

Link[11, NonAlternating, 113], Link[11, NonAlternating, 114],

Link[11, NonAlternating, 115]}</nowiki>}}

Thus, there are 21 links in the Knot Atlas that have a non-prime component. The first eight of those are:

{{Knot Image Quadruple|L10a38|gif|L10a39|gif|L10a46|gif|L10n35|gif}}
{{Knot Image Quadruple|L10n36|gif|L10n37|gif|L10n38|gif|L10n39|gif}}

IdentifyWithin.m

2013-10-20T18:54:31Z

Drorbn:

(*

The program <code>IdentifyWithin</code>, documented in the page on [[Identifying Knots within a List]]. It is not part of the package <code>KnotTheory`</code> but is designed to work with it. Use <code><nowiki>Import["http://katlas.org/w/index.php?title=IdentifyWithin.m&action=raw"]</nowiki></code> to download into a mathematica session, or copy-paste the text below ignoring the <code>*)</code> on the first line and the <code>(*</code> on the last.
<pre>
*)
Options[IdentifyWithin] = {
UseInvariants -> {Jones[#][q] &, HOMFLYPT[#][a, z] &,
Kauffman[#][a, z] &},
ConnectedSum -> "True"};
IdentifyWithin[L_, H_List, opts___Rule] :=

Module[
{div, j = 1, l, i = 1, u, mu, t, mt, out = {}, out1 = {}, nk, mnk,
mnk1, p, mp, m, p1,
invariants = (UseInvariants /. {opts} /. Options[IdentifyWithin]),
connectedsum = (ConnectedSum /. {opts} /.
Options[IdentifyWithin])},

NormalizeP[poly_] := Module[{t1, i1},
(For[i1 = 1 ; t1 := FactorList[poly],
i1 <= Length[Variables[poly]], i1++,
t1 =
DeleteCases[t1, {Variables[poly][[i1]], _Integer} | {1, 1}]];
Times @@ Power @@@ t1 )];

l := Length[invariants];
u[0] = mu[0] = H;
While[i <= l && ! Length[out] === 1,
t[i] = invariants[[i]][L];
mt[i] = invariants[[i]][Mirror[L]];
u[i] = Select[u[i - 1], t[i] == invariants[[i]][#] &];
mu[i] = Select[mu[i - 1], mt[i] == invariants[[i]][#] &];
out = Flatten[{u[i], Mirror /@ mu[i]}];
i++];

Which[
Length[out] >= 2, DeleteCases[out, Mirror[Knot[0, 1]]],
Length[out] == 1,
out = If[u[i - 1] != {}, u[i - 1], Mirror /@ mu[i - 1]],
connectedsum === "True", i = 1; nk[0] = mnk[0] = H;

While[Length[out1] != 1 && i <= l,
p[i] = NormalizeP[t[i]];
mp[i] = NormalizeP[mt[i]];
nk[i] =
Select[nk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3;
PolynomialRemainder[p[i], p1, Variables[p[i]][[1]]] ===
0 ) &];
mnk[i] =
Select[mnk[i - 1], (p1 = NormalizeP[invariants[[i]][#]]; z = 3;
PolynomialRemainder[mp[i], p1, Variables[p[i]][[1]]] ===
0 ) &];

Clear[z];

mnk1[i] = Mirror /@ mnk[i];
div = Flatten[{nk[i], mnk1[i]}];
div = DeleteCases[div, Knot[0, 1] | Mirror[Knot[0, 1]]];

If[div == {}, out1 = {},
For[m = 1;
W[0] =
CS[0] = Select[
Flatten /@ Flatten[Outer[List, div, div, 1], 1], OrderedQ],
Length[W[m - 1][[1]]] < 4, m++,
W[m] = Select[
Flatten /@ Flatten[Outer[List, div, W[m - 1], 1], 1],
OrderedQ];
CS[m] = Flatten[{CS[m - 1], W[m]}, 1];
];
out1 =
Select[CS[m - 1],
Expand[Times @@ invariants[[i]] /@ #] == t[i] &];
];
i++];
If[out1 == {}, {}, ConnectedSum @@@ out1], True, {}
]
];
(* </pre> *)

Template:Knot Image Quadruple

2013-08-30T00:01:42Z

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{| align=center
|- align=center
|[[Image:{{{1}}}.{{{2}}}|120px|link={{{1}}}]] [[{{{1}}}]]
|[[Image:{{{3}}}.{{{4}}}|120px|link={{{3}}}]] [[{{{3}}}]]
|[[Image:{{{5}}}.{{{6}}}|120px|link={{{5}}}]] [[{{{5}}}]]
|[[Image:{{{7}}}.{{{8}}}|120px|link={{{7}}}]] [[{{{7}}}]]
|}

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{| align=center
|- align=center
|[[Image:{{{1}}}.{{{2}}}|180px|link={{{1}}}]] [[{{{1}}}]]
|[[Image:{{{3}}}.{{{4}}}|180px|link={{{3}}}]] [[{{{3}}}]]
|[[Image:{{{5}}}.{{{6}}}|180px|link={{{5}}}]] [[{{{5}}}]]
|}
[[Image:{{{5}}}.{{{6}}}|120px]]

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The Multivariable Alexander Polynomial

2013-08-29T23:56:59Z

Drorbn: /* Links with Vanishing Multivariable Alexander Polynomial */

{{Manual TOC Sidebar}}

{{Startup Note}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>MultivariableAlexander</nowiki> |
out= <nowiki>MultivariableAlexander[L][t] returns the multivariable Alexander polynomial
of a link L as a function of the variable t[1], t[2], ..., t[c], where c
is the number of components of L. MultivariableAlexander[L, Program -> prog][t]
uses the program prog to perform the computation. The currently available
programs are "MVA1", written by Dan Carney in Toronto in the summer of 2005,
and the faster "MVA2" (default), written by Jana Archibald in Toronto in 2008-9.</nowiki> |
about= <nowiki>The multivariable Alexander program "MVA1" was
written by Dan Carney at the University of Toronto in the summer of 2005; "MVA2"
was written by Jana Archibald in Toronto in 2008-9.</nowiki>}}


{{Knot Image|L8a21|gif}}

The link [[L8a21]] is symmetric under cyclic permutations of its components but not under interchanging two adjacent components. It is amusing to see how this is reflected in its multivariable Alexander polynomial:



{{InOut|
n = 3 |
in = <nowiki>mva = MultivariableAlexander[Link[8, Alternating, 21]][t] /. {
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
}</nowiki> |
out= <nowiki>(-t1 - t2 + t1 t2 - t3 + 2 t1 t3 + t2 t3 - t1 t2 t3 - t4 + t1 t4 +

2 t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4) /

(Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])</nowiki>}}




{{InOut|
n = 4 |
in = <nowiki>mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})</nowiki> |
out= <nowiki>0</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>Simplify[mva - (mva /. {t1->t2, t2->t1})]</nowiki> |
out= <nowiki> (t1 - t2) (t3 - t4)
-----------------------------------
Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4]</nowiki>}}


But notice the funny labelling of the components! The program <code>MultivariableAlexander</code> orders the variables in its output (typically denoted <code>t[i]</code>) in the same order as the order of the components of a link <code>L</code> as they appear within <code>Skeleton[L]</code>. Hence we had to rename <code>t[3]</code> to be <code>t4</code> and <code>t[4]</code> to be <code>t3</code>.

====Links with Vanishing Multivariable Alexander Polynomial====

There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is <math>0</math>. Here they are:



{{InOut|
n = 6 |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]</nowiki> |
out= <nowiki>{Link[9, NonAlternating, 27], Link[10, NonAlternating, 32],

Link[10, NonAlternating, 36], Link[10, NonAlternating, 107],

Link[11, NonAlternating, 244], Link[11, NonAlternating, 247],

Link[11, NonAlternating, 334], Link[11, NonAlternating, 381],

Link[11, NonAlternating, 396], Link[11, NonAlternating, 404],

Link[11, NonAlternating, 406]}</nowiki>}}


{{Knot Image Quadruple|L9n27|gif|L10n32|gif|L10n36|gif|L10n107|gif}}
{{Knot Image Quadruple|L11n244|gif|L11n247|gif|L11n334|gif|L11n381|gif}}
{{Knot Image Triple|L11n396|gif|L11n404|gif|L11n406|gif}}

[[User:Drorbn|Dror]] doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots. (Though see the [[Talk:The Multivariable Alexander Polynomial|Talk Page]]).

====Detecting a Link Using the Multivariable Alexander Polynomial====

[[Image:Celtic-knot-basic-alternate.gif|thumb|right|200px|A mystery link]] On May 1, 2007 [[User:AnonMoos|AnonMoos]] asked [[User:Drorbn|Dror]] if he could identify the link in the figure on the right. So Dror typed:



{{InOut|
n = 7 |
in = <nowiki>mva = MultivariableAlexander[L = PD[
X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9],
X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3],
X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10],
X[19, 4, 20, 5], X[21, 7, 22, 6]
]][t]</nowiki> |
out= <nowiki> 2
-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -

2 2 2 2 2
2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /

3/2 3/2
(t[1] t[2] ))</nowiki>}}


We don't know whether our mystery link appears in the link table as is, or as a mirror, or with its two components switched. Hence we let <code>AllPossibilities</code> contain the multivariable Alexander polynomials of all those possibilities:



{{InOut|
n = 8 |
in = <nowiki>AllPossibilities = Union[Flatten[
{mva, -mva} /. {{}, {t[1] -> t[2], t[2] -> t[1]}}
]]</nowiki> |
out= <nowiki> 2
{-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] +

2 2 2 2 2
2 t[1] t[2] - 2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2]

3/2 3/2
)) / (t[1] t[2] )),

2
((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -

2 2 2 2 2
2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /

3/2 3/2
(t[1] t[2] )}</nowiki>}}


Finally, let us locate our link in the link table:



{{InOut|
n = 9 |
in = <nowiki>Select[
AllLinks[],
MemberQ[AllPossibilities, MultivariableAlexander[#][t]] &
]</nowiki> |
out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}


And just to be sure,



{{InOut|
n = 10 |
in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> |
out= <nowiki> -(17/2) 4 8 12 16 18 17 15
{q - ----- + ----- - ----- + ---- - ---- + ---- - ---- +
15/2 13/2 11/2 9/2 7/2 5/2 3/2
q q q q q q q

10 3/2 5/2
------- - 7 Sqrt[q] + 3 q - q ,
Sqrt[q]

-(5/2) 3 7 3/2 5/2
-q + ---- - ------- + 10 Sqrt[q] - 15 q + 17 q -
3/2 Sqrt[q]
q

7/2 9/2 11/2 13/2 15/2 17/2
18 q + 16 q - 12 q + 8 q - 4 q + q }</nowiki>}}


Thus the mystery link is the mirror image of [[L11a289]].

The Multivariable Alexander Polynomial

2013-08-29T23:56:19Z

Drorbn: /* Links with Vanishing Multivariable Alexander Polynomial */

{{Manual TOC Sidebar}}

{{Startup Note}}



{{HelpAndAbout|
n = 1 |
n1 = 2 |
in = <nowiki>MultivariableAlexander</nowiki> |
out= <nowiki>MultivariableAlexander[L][t] returns the multivariable Alexander polynomial
of a link L as a function of the variable t[1], t[2], ..., t[c], where c
is the number of components of L. MultivariableAlexander[L, Program -> prog][t]
uses the program prog to perform the computation. The currently available
programs are "MVA1", written by Dan Carney in Toronto in the summer of 2005,
and the faster "MVA2" (default), written by Jana Archibald in Toronto in 2008-9.</nowiki> |
about= <nowiki>The multivariable Alexander program "MVA1" was
written by Dan Carney at the University of Toronto in the summer of 2005; "MVA2"
was written by Jana Archibald in Toronto in 2008-9.</nowiki>}}


{{Knot Image|L8a21|gif}}

The link [[L8a21]] is symmetric under cyclic permutations of its components but not under interchanging two adjacent components. It is amusing to see how this is reflected in its multivariable Alexander polynomial:



{{InOut|
n = 3 |
in = <nowiki>mva = MultivariableAlexander[Link[8, Alternating, 21]][t] /. {
t[1] -> t1, t[2] -> t2, t[3] -> t4, t[4] -> t3
}</nowiki> |
out= <nowiki>(-t1 - t2 + t1 t2 - t3 + 2 t1 t3 + t2 t3 - t1 t2 t3 - t4 + t1 t4 +

2 t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4) /

(Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])</nowiki>}}




{{InOut|
n = 4 |
in = <nowiki>mva - (mva /. {t1->t2, t2->t3, t3->t4, t4->t1})</nowiki> |
out= <nowiki>0</nowiki>}}




{{InOut|
n = 5 |
in = <nowiki>Simplify[mva - (mva /. {t1->t2, t2->t1})]</nowiki> |
out= <nowiki> (t1 - t2) (t3 - t4)
-----------------------------------
Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4]</nowiki>}}


But notice the funny labelling of the components! The program <code>MultivariableAlexander</code> orders the variables in its output (typically denoted <code>t[i]</code>) in the same order as the order of the components of a link <code>L</code> as they appear within <code>Skeleton[L]</code>. Hence we had to rename <code>t[3]</code> to be <code>t4</code> and <code>t[4]</code> to be <code>t3</code>.

====Links with Vanishing Multivariable Alexander Polynomial====

There are 11 links with up to 11 crossings whose multivariable Alexander polynomial is <math>0</math>. Here they are:



{{InOut|
n = 6 |
in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == 0) &]</nowiki> |
out= <nowiki>{Link[9, NonAlternating, 27], Link[10, NonAlternating, 32],

Link[10, NonAlternating, 36], Link[10, NonAlternating, 107],

Link[11, NonAlternating, 244], Link[11, NonAlternating, 247],

Link[11, NonAlternating, 334], Link[11, NonAlternating, 381],

Link[11, NonAlternating, 396], Link[11, NonAlternating, 404],

Link[11, NonAlternating, 406]}</nowiki>}}


{{Knot Image Quadruple|L9n27|gif|L10n32|gif|L10n36|gif|L10n107|gif}}
{{Knot Image Quadruple|L11n244|gif|L11n247|gif|L11n334|gif|L11n381|gif}}
{{Knot Image Pair|L11n396|gif|L11n404|gif|L11n406|gif}}

[[User:Drorbn|Dror]] doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots. (Though see the [[Talk:The Multivariable Alexander Polynomial|Talk Page]]).

====Detecting a Link Using the Multivariable Alexander Polynomial====

[[Image:Celtic-knot-basic-alternate.gif|thumb|right|200px|A mystery link]] On May 1, 2007 [[User:AnonMoos|AnonMoos]] asked [[User:Drorbn|Dror]] if he could identify the link in the figure on the right. So Dror typed:



{{InOut|
n = 7 |
in = <nowiki>mva = MultivariableAlexander[L = PD[
X[1, 16, 2, 17], X[3, 15, 4, 14], X[5, 8, 6, 9],
X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3],
X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10],
X[19, 4, 20, 5], X[21, 7, 22, 6]
]][t]</nowiki> |
out= <nowiki> 2
-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -

2 2 2 2 2
2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /

3/2 3/2
(t[1] t[2] ))</nowiki>}}


We don't know whether our mystery link appears in the link table as is, or as a mirror, or with its two components switched. Hence we let <code>AllPossibilities</code> contain the multivariable Alexander polynomials of all those possibilities:



{{InOut|
n = 8 |
in = <nowiki>AllPossibilities = Union[Flatten[
{mva, -mva} /. {{}, {t[1] -> t[2], t[2] -> t[1]}}
]]</nowiki> |
out= <nowiki> 2
{-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] +

2 2 2 2 2
2 t[1] t[2] - 2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2]

3/2 3/2
)) / (t[1] t[2] )),

2
((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -

2 2 2 2 2
2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /

3/2 3/2
(t[1] t[2] )}</nowiki>}}


Finally, let us locate our link in the link table:



{{InOut|
n = 9 |
in = <nowiki>Select[
AllLinks[],
MemberQ[AllPossibilities, MultivariableAlexander[#][t]] &
]</nowiki> |
out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}


And just to be sure,



{{InOut|
n = 10 |
in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> |
out= <nowiki> -(17/2) 4 8 12 16 18 17 15
{q - ----- + ----- - ----- + ---- - ---- + ---- - ---- +
15/2 13/2 11/2 9/2 7/2 5/2 3/2
q q q q q q q

10 3/2 5/2
------- - 7 Sqrt[q] + 3 q - q ,
Sqrt[q]

-(5/2) 3 7 3/2 5/2
-q + ---- - ------- + 10 Sqrt[q] - 15 q + 17 q -
3/2 Sqrt[q]
q

7/2 9/2 11/2 13/2 15/2 17/2
18 q + 16 q - 12 q + 8 q - 4 q + q }</nowiki>}}


Thus the mystery link is the mirror image of [[L11a289]].

Template:Knot Image Triple

2013-08-29T23:49:40Z

Drorbn:

Template:Knot Image

2013-08-29T23:48:20Z

Drorbn:

{| align=center
|- align=center
|[[Image:{{{1}}}.{{{2}}}|180px|link={{{1}}}]] [[{{{1}}}]]
|}

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2013-08-29T23:47:22Z

Drorbn:

{| align=center
|- align=center
|[[Image:{{{1}}}.{{{2}}}|180px|link={{{1}}}]] [[{{{1}}}]]
|[[Image:{{{3}}}.{{{4}}}|180px|link={{{3}}}]] [[{{{3}}}]]
|}