Knot Atlas - User contributions [en]

Using the LinKnot package

2006-04-07T12:56:49Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip download LinKnot.zip] from the [http://www.mi.sanu.ac.yu/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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the directory to LinKnot, add the path to KnotTheory, and from any nb-file run:

SetDirectory["LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if the directories LinKnot and KnotTheory are both at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Using the LinKnot package

2006-04-07T12:52:57Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip download] LinKnot.zip from the [http://www.mi.sanu.ac.yu/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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the directory to LinKnot, add the path to KnotTheory, and from any nb-file run:

SetDirectory["LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if the directories KnotTheory and LinKnot are both at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Using the LinKnot package

2006-04-07T12:51:52Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the directory to LinKnot, add the path to KnotTheory, and from any nb-file run:

SetDirectory["LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if the directories KnotTheory and LinKnot are both at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Conway Notation

2006-04-07T12:50:21Z

Jablans:

{{Manual TOC Sidebar}}

====Conway notation and <tt>KnotTheory`</tt>====

<tt>KnotTheory`</tt> understands the Conway notation for knots and links (see {{ref|Conway}} and down below), though the conversion
between Conway notation and other knot presentations known to <tt>KnotTheory`</tt> (a necessary first step for using most of the KnotTheory` functionnality) requires the packages ''K2K'' (KNOT 2000, by M.Ochiai and N.Imafuji) and ''LinKnot'' (by S. Jablan and R. Sazdanovic). For the download and installation of the ''LinKnot'' package see [[Using the LinKnot package]].

After the installation, set the directory to LinKnot, add the path to KnotTheory, and from any nb-file run:

SetDirectory["LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if the directories KnotTheory and LinKnot are both at C: run

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to see the complete description of the program ''LinKnot'', open the file K2KL.nb from the directory LinKnot and run the programs in the same way.



{{HelpAndAbout|
n = 3 |
n1 = 4 |
in = <nowiki>ConwayNotation</nowiki> |
out= <nowiki>ConwayNotation[s] represents the knot or link whose Conway notation is the string s. ConwayNotation[K], where K is a knot or a link with up to 12 crossings, returns ConwayNotation[s], where s is a string containing the Conway notation of K.</nowiki> |
about= <nowiki>The program ConwayNotation relies on code from the LinKnot package by Slavik Jablan and Ramila Sazdanovic.</nowiki>}}


A well known example of a knot with an Alexander polynomial equal to the Alexander polynomial of the unknot is the (-3,5,7)-pretzel knot <math>K</math>. Let us verify that, check (using the Jones polynomial that <math>K</math> is not the unknot and find a (rather unattractive) braid whose closure is <math>K</math>:



{{Graphics|
n = 6 |
in = <nowiki>DrawMorseLink[K = ConwayNotation["-3,5,7"]] // Show</nowiki> |
img= Conway_Notation_Out_5.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 7 |
in = <nowiki>Alexander[K][t]</nowiki> |
out= <nowiki>1</nowiki>}}




{{InOut|
n = 8 |
in = <nowiki>Jones[K][q]</nowiki> |
out= <nowiki> -12 -11 -10 2 -8 -7 -5 -4 2 -2 1
q - q + q - -- + q - q + q - q + -- - q + -
9 3 q
q q</nowiki>}}




{{InOut|
n = 9 |
in = <nowiki>br = BR[K]</nowiki> |
out= <nowiki>BR[14, {1, 2, 3, -4, -5, -6, -7, 8, -7, 6, 5, 4, -3, -2, -1, -6, -5,

-4, -3, -2, 9, 8, 7, 6, -5, 4, -3, 7, -8, -7, -9, -8, 10, 9, -8,

-11, -10, 12, 11, -10, 9, -8, -13, -12, -11, 10, 9, -8, -7, 6, -5,

4, -5, -7, 8, -7, -6, -7, -9, 8, -7, 6, 5, -4, 3, 2, -6, -7, -10,

-9, 11, 10, -9, 8, -7, 6, 5, -4, 3, -6, 5, 4, -6, 5, 7, 6, -7, -8,

9, 8, -7, 12, -11, 10, -9, 13, -12, 11, -10}]</nowiki>}}




{{Graphics|
n = 11 |
in = <nowiki>BraidPlot[br] // Show</nowiki> |
img= Conway_Notation_Out_10.gif |
out= <nowiki>-Graphics-</nowiki>}}


====Some generalities about the Conway notation====

Conway notation was introduced by J.H. Conway in 1967 (see {{ref|Conway}}). The main building blocks for the Conway notation are 4-tangles. A 4-''tangle'' in a knot or link projection is a region in the projection plane <math>{\mathbb R}^2</math> (or on the sphere <math>S^3</math>) surrounded with a circle such that the projection intersects with the circle exactly four times. The elementary tangles are:
<table cellspacing=0 cellpadding=0 border=0 align=center>
<tr><td>[[Image:tangle0.jpg]]</td><td>[[Image:tangle1.jpg]]</td>
<td>[[Image:tangle-1.jpg]]</td></tr>
</table>

Tangles could be combined and modified by one unary operation <math>a\mapsto-a</math> and three binary operations: ''sum'', ''product'' and ''ramification'', taking tangles <math>a</math>, <math>b</math> to new tangles <math>a+b</math>, <math>a\,b</math> and <math>a,b</math>. Here <math>-a</math> is the image of <math>a</math> under reflection in the NW-SE mirror line, <math>a+b</math> is obtained by placing <math>a</math> and <math>b</math> side by side with <math>a</math> on the left and <math>b</math> on the right. <math>a\,b</math> is simply <math>(-a)+b</math>, and finally, <math>a,b = (-a)+(-b)</math>.

<table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:033.jpg]]</td><td>[[Image:ConwayRamification.jpg]]</td></tr>
<tr><td align=center>Sum and product of tangles</td><td align=center>Ramification of tangles</td></tr>
</table>

A ''rational tangle'' is any tangle obtained from the elementary tangles using only the operation of product. A ''rational knot'' or a ''rational link'' is the numerator closure of a rational tangle. A knot or link is called ''algebraic'' if it can be obtained as the closure of a tangle obtained from rational tangles using the operations above.

Knot or links that can not be obtained in this way are called ''non-algebraic''. They can all be obtained in the following manner: start with a ''basic polyhedron'' <math>P</math>, a 4-valent graph without digons, with vertices numbered <math>1</math> through <math>n</math>. Now substitute tangles <math>t_1</math> through <math>t_n</math> into these vertices.

The Conway notation for such knots and links consists of the symbol <math>ni^\star</math> of a basic polyhedron <math>P</math> where <math>n</math> is the number of vertices and <math>i</math> is the index of <math>P</math> in some fixed list of basic polyhedra with <math>n</math> vertices, followed by the symbols for the tangles <math>t_1</math> through <math>t_n</math> separated by dots.

For example, the knot [[4_1]] is denoted by "2 2", the knot [[9_5]] by "5 1 3", the link [[L5a1]] is denoted by "2 1 2", the link [[L9a24]] by "3 1,3,2" (all of them contain spaces between tangles), etc. A sequence of k pluses at the end of Conway symbol is denoted by +k, and the sequence of k minuses by +-k (e.g., knot [[10_76]] given in Conway notation as 3,3,2++ is denoted by "3,3,2+2", and the mirror of the link [[L9n21]] whose Conway notation is 3,2,2,2-- is given by "3,2,2,2+-2"). The space is used in the same way in all other symbols.

For the basic polyhedra with <math>N<10</math> crossings the standard notation is used (.1 , 6*, 8*, 9*, where the symbol for 6* can be ommitted). For example, the knot [[10_95]] is denoted by ".2 1 0.2.2", and [[10_101]] by "2 1..2..2". For higher values of <math>N</math> it is used notation in which the first number is the number of crossings, and the next is the ordering number of polyhedron (e.g., 101*, 102*, 103* for <math>N=10</math> denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for <math>N=11</math> denoting 11*, 11**, 11***, respectively, ''etc''.).

The order of basic polyhedra for <math>N=12</math> corresponds to the list in {{ref|Caudron}}, so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For <math>N>12</math> the database of basic polyhedra is produced from the list of simple 4-regular 4-edge-connected but not 3-connected plane graphs generated by Brendan McKay using the program "plantri" written by Gunnar Brinkmann and Brendan McKay (http://cs.anu.edu.au/~bdm/plantri/). PolyBase.m is automatically downloaded and it cointains basic polyhedra up to 16 crossings. In order to work with the basic polyhedra up tp 20 vertices, one needs to open an additional database PolyBaseN.m, for <math>N=17</math> to <math>N=20</math> (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for <math>N=17</math>).

Note: Together with the classical notation, Conway symbols are given in the book ''Knots and Links'' by D.~Rolfsen. However if you try to draw some knots or links from their Conway symbols the obtained projection might be non-isomorphic with the one given in Rolfsen, for example knot [[9_15]] denoted in Conway notation as 2 3 2 2 gives projection with 5, and not 4 digons.

{{note|Caudron}} A. Caudron, ''Classification des noeuds et des enlancements''. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.

{{note|Conway}} J. H. Conway, ''An Enumeration of Knots and Links, and Some of Their Algebraic Properties.'' In Computation Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press, pp. 329-358, 1967.

Using the LinKnot package

2006-04-07T12:45:08Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the directory to LinKnot, add the path to ''KnotTheory`'', and from any nb-file run:

SetDirectory["LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if the directories KnotTheory and LinKnot are both at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Using the LinKnot package

2006-04-07T12:44:31Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the directory to LinKnot, add the path to ''KnotTheory`'', and from any nb-file run:

SetDirectory["LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if KnotTheory and LinKnot are both at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Using the LinKnot package

2006-04-07T12:43:58Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the directory to LinKnot, add the path to ''KnotTheory`'', and from any nb-file run:

SetDirectory["Path to LinKnot Directory"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if KnotTheory and LinKnot are both at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Using the LinKnot package

2006-04-07T12:38:43Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the path to ''KnotTheory`'' and from any nb-file run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if ''KnotTheory'' is also at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or S.Jablan (jablans@yahoo.com) .

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

Using the LinKnot package

2006-04-07T12:33:03Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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 links, and a graphical interface for drawing knots and links by hand.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the path to ''KnotTheory`'' and from any nb-file run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if ''KnotTheory'' is also at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or Slavik Jablan (jablans@yahoo.com).

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

Using the LinKnot package

2006-04-07T12:31:51Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/vismath/linknot/LinKnot.zip 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:

1) extract LinKnot.zip anywhere (e.g., to the local disc C:). It will automatically create new folder LinKnot

2) Set the path to ''KnotTheory`'' and from any nb-file run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "Path to KnotTheory"];

<< KnotTheory`

For example, if ''KnotTheory'' is also at C: run:

SetDirectory["C:\\LinKnot"];

<< LinKnots`

AppendTo[$Path, "C:\\"];

<< KnotTheory`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run together ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory LinKnot 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 [[User:Scott|Scott]] or Slavik Jablan (jablans@yahoo.com).

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

Using the LinKnot package

2006-04-05T12:46:46Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/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:

1) extract it in the Mathematica directory

C:\ProgramFiles\WolframResearch\Mathematica\5.2

It will automatically create new folder LinKnot

2) from any nb-file run:

SetDirectory["LinKnot"];

<<LinKnots`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run the both programs ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory

C://ProgramFiles/WolframResearch/Mathematica/5.2/LinKnot

and run

SetDirectory["LinKnot"];

<<LinKnots`

The directory LinKnot contains ''Knot Theory'' package, that we will try to keep updated. However, if you have in your computer the more recent version of ''Knot Theory'', you can run:

SetDirectory["LinKnot"];

<<LinKnots`

AppendTo[$Path, \\Path to KnotTheory];

Get["Path to Knot Theory\\init.m"]

of just update the version of ''KnotTheory'' already existing in the directory LinKnot by extracting the most recent file KnotTheory.zip into the directory LinKnot.

Finally, ''LinKnot'' and Knot Theory can be placed anywhere you like, but in this case you need to set the paths by:

SetDirectory["Path to LinKnot"];

<<LinKnots`

AppendTo[$Path,"Path to Knot Theory"];

Get["Path to KnotTheory\\KnotTheory\\init.m"]

After you've done this, everything should just work. If everything doesn't just work, please complain to [[User:Scott|Scott]] or Slavik Jablan (jablans@yahoo.com).

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

Using the LinKnot package

2006-04-05T12:45:28Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/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:

1) extract it in the Mathematica directory

C:\ProgramFiles\WolframResearch\Mathematica\5.2

It will automatically create new folder LinKnot

2) from any nb-file run:

SetDirectory["LinKnot"];

<<LinKnots`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run the both programs ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory

C://ProgramFiles/WolframResearch/Mathematica/5.2/LinKnot

and run

SetDirectory["LinKnot"];

<<LinKnots`

The directory LinKnot contains ''Knot Theory'' package, that we will try to keep updated. However, if you have in your computer the more recent version of ''Knot Theory'', you can run:

SetDirectory["LinKnot"];

<<LinKnots`

AppendTo[$Path, \\Path to KnotTheory];

Get["Path to Knot Theory\\init.m"]

of just update the version of ''KnotTheory'' already existing in the directory LinKnot by extracting the file most recent file KnotTheory.zip into the directory LinKnot.

Finally, ''LinKnot'' and Knot Theory can be placed anywhere you like, but in this case you need to set the paths by:

SetDirectory["Path to LinKnot"];

<<LinKnots`

AppendTo[$Path,"Path to Knot Theory"];

Get["Path to KnotTheory\\KnotTheory\\init.m"]

After you've done this, everything should just work. If everything doesn't just work, please complain to [[User:Scott|Scott]] or Slavik Jablan (jablans@yahoo.com).

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

Using the LinKnot package

2006-04-05T12:41:54Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/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:

1) extract it in the Mathematica directory

C:\ProgramFiles\WolframResearch\Mathematica\5.2

It will automatically create new folder LinKnot

2) from any nb-file run:

SetDirectory["LinKnot"];

<<LinKnots`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run the both programs ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory

C://ProgramFiles/WolframResearch/Mathematica/5.2/LinKnot

and run

SetDirectory["LinKnot"];

<<LinKnots`

The directory LinKnot contains ''Knot Theory'' package, that we will try to keep updated. However, if you have in your computer the more recent version of ''Knot Theory'', you can run:

SetDirectory["LinKnot"];

<<LinKnots`

AppendTo[$Path, \\Path to KnotTheory];

Get["Path to Knot Theory\\init.m"]

of just update the version of ''KnotTheory'' already existing in the directory LinKnot by extracting the file most recent file KnotTheory.zip into the directory LinKnot.

Finally, ''LinKnot'' and Knot Theory can be placed anywhere you like, but in this case you need to set the paths by:

SetDirectory["Path to LinKnot"];

<<LinKnots`

AppendTo[$Path,"Path to Knot Theory"];

Get["Path to Knot Theory\\KnotTheory\\init.m"]

After you've done this, everything should just work. If everything doesn't just work, please complain to [[User:Scott|Scott]] or Slavik Jablan (jablans@yahoo.com).

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

Using the LinKnot package

2006-04-05T12:40:53Z

Jablans:

{{Manual TOC Sidebar}}

The Mathematica package [http://www.mi.sanu.ac.yu/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. The package ''LinKnot'' 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.

To use ''LinKnot'' with ''KnotTheory'', you should download LinKnot.zip from the [http://www.mi.sanu.ac.yu/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:

1) extract it in the Mathematica directory

C:\ProgramFiles\WolframResearch\Mathematica\5.2

It will automatically create new folder LinKnot

2) from any nb-file run:

SetDirectory["LinKnot"];

<<LinKnots`

Then you can work with the both programs ''KnotTheory'' and ''LinKnot''.

If you like to have the complete program ''LinKnot'' visible (with the usage, description of all ''LinKnot'' functions, ''etc''.) and run the both programs ''KnotTheory'' and ''LinKnot'', you can open the file K2KL.nb from the directory

C://ProgramFiles/WolframResearch/Mathematica/5.2/LinKnot

and run

SetDirectory["LinKnot"];

<<LinKnots`

The directory LinKnot contains ''Knot Theory'' package, that we will try to keep updated. However, if you have in your computer the more recent version of ''Knot Theory'', you can run:

SetDirectory["LinKnot"];

<<LinKnots`

AppendTo[$Path, \\Path to KnotTheory];

Get["Path to Knot Theory\\init.m"]

of just update the version of ''KnotTheory'' already existing in the directory LinKnot by extracting the file most recent file KnotTheory.zip into the directory LinKnot.

Finally, ''LinKnot'' and Knot Theory can be placed anywhere you like, but in this case you need to set the paths by:

SetDirectory["Path to LinKnot"];

<<LinKnots`

AppendTo[$Path,"Path to Knot Theory"];

Get["Path to Knot Theory\\KnotTheory\\init.m"]

After you've done this, everything should just work. If everything doesn't just work, please complain to [[User:Scott|Scott]].

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

Conway Notation

2006-04-05T11:23:25Z

Jablans:

{{Manual TOC Sidebar}}

====Conway notation and <tt>KnotTheory`</tt>====

<tt>KnotTheory`</tt> understands the Conway notation for knots and links (see {{ref|Conway}} and down below), though the conversion
between Conway notation and other knot presentations known to <tt>KnotTheory`</tt> (a necessary first step for using most of the KnotTheory` functionnality) requires the packages ''K2K'' (KNOT 2000, by M.Ochiai and N.Imafuji) and ''LinKnot'' (by S. Jablan and R. Sazdanovic). For the download and installation of the ''LinKnot'' package see [[Using the LinKnot package]].

After the installation, from any nb-file run:

SetDirectory["LinKnot"]

<<LinKnots`

After that, you can work in the both programs ''KnotTheory'' and ''LinKnot''.

If you like to see the complete description of the program ''LinKnot'', open the file K2KL.nb from the directory LinKnot and run it.



{{HelpAndAbout|
n = 3 |
n1 = 4 |
in = <nowiki>ConwayNotation</nowiki> |
out= <nowiki>ConwayNotation[s] represents the knot or link whose Conway notation is the string s. ConwayNotation[K], where K is a knot or a link with up to 12 crossings, returns ConwayNotation[s], where s is a string containing the Conway notation of K.</nowiki> |
about= <nowiki>The program ConwayNotation relies on code from the LinKnot package by Slavik Jablan and Ramila Sazdanovic.</nowiki>}}


A well known example of a knot with an Alexander polynomial equal to the Alexander polynomial of the unknot is the (-3,5,7)-pretzel knot <math>K</math>. Let us verify that, check (using the Jones polynomial that <math>K</math> is not the unknot and find a (rather unattractive) braid whose closure is <math>K</math>:



{{Graphics|
n = 6 |
in = <nowiki>DrawMorseLink[K = ConwayNotation["-3,5,7"]] // Show</nowiki> |
img= Conway_Notation_Out_5.gif |
out= <nowiki>-Graphics-</nowiki>}}




{{InOut|
n = 7 |
in = <nowiki>Alexander[K][t]</nowiki> |
out= <nowiki>1</nowiki>}}




{{InOut|
n = 8 |
in = <nowiki>Jones[K][q]</nowiki> |
out= <nowiki> -12 -11 -10 2 -8 -7 -5 -4 2 -2 1
q - q + q - -- + q - q + q - q + -- - q + -
9 3 q
q q</nowiki>}}




{{InOut|
n = 9 |
in = <nowiki>br = BR[K]</nowiki> |
out= <nowiki>BR[14, {1, 2, 3, -4, -5, -6, -7, 8, -7, 6, 5, 4, -3, -2, -1, -6, -5,

-4, -3, -2, 9, 8, 7, 6, -5, 4, -3, 7, -8, -7, -9, -8, 10, 9, -8,

-11, -10, 12, 11, -10, 9, -8, -13, -12, -11, 10, 9, -8, -7, 6, -5,

4, -5, -7, 8, -7, -6, -7, -9, 8, -7, 6, 5, -4, 3, 2, -6, -7, -10,

-9, 11, 10, -9, 8, -7, 6, 5, -4, 3, -6, 5, 4, -6, 5, 7, 6, -7, -8,

9, 8, -7, 12, -11, 10, -9, 13, -12, 11, -10}]</nowiki>}}




{{Graphics|
n = 11 |
in = <nowiki>BraidPlot[br] // Show</nowiki> |
img= Conway_Notation_Out_10.gif |
out= <nowiki>-Graphics-</nowiki>}}


====Some generalities about the Conway notation====

Conway notation was introduced by J.H. Conway in 1967 (see {{ref|Conway}}). The main building blocks for the Conway notation are 4-tangles. A 4-''tangle'' in a knot or link projection is a region in the projection plane <math>{\mathbb R}^2</math> (or on the sphere <math>S^3</math>) surrounded with a circle such that the projection intersects with the circle exactly four times. The elementary tangles are:
<table cellspacing=0 cellpadding=0 border=0 align=center>
<tr><td>[[Image:tangle0.jpg]]</td><td>[[Image:tangle1.jpg]]</td>
<td>[[Image:tangle-1.jpg]]</td></tr>
</table>

Tangles could be combined and modified by one unary operation <math>a\mapsto-a</math> and three binary operations: ''sum'', ''product'' and ''ramification'', taking tangles <math>a</math>, <math>b</math> to new tangles <math>a+b</math>, <math>a\,b</math> and <math>a,b</math>. Here <math>-a</math> is the image of <math>a</math> under reflection in the NW-SE mirror line, <math>a+b</math> is obtained by placing <math>a</math> and <math>b</math> side by side with <math>a</math> on the left and <math>b</math> on the right. <math>a\,b</math> is simply <math>(-a)+b</math>, and finally, <math>a,b = (-a)+(-b)</math>.

<table cellspacing=0 cellpadding=0 border=0>
<tr><td>[[Image:033.jpg]]</td><td>[[Image:ConwayRamification.jpg]]</td></tr>
<tr><td align=center>Sum and product of tangles</td><td align=center>Ramification of tangles</td></tr>
</table>

A ''rational tangle'' is any tangle obtained from the elementary tangles using only the operation of product. A ''rational knot'' or a ''rational link'' is the numerator closure of a rational tangle. A knot or link is called ''algebraic'' if it can be obtained as the closure of a tangle obtained from rational tangles using the operations above.

Knot or links that can not be obtained in this way are called ''non-algebraic''. They can all be obtained in the following manner: start with a ''basic polyhedron'' <math>P</math>, a 4-valent graph without digons, with vertices numbered <math>1</math> through <math>n</math>. Now substitute tangles <math>t_1</math> through <math>t_n</math> into these vertices.

The Conway notation for such knots and links consists of the symbol <math>ni^\star</math> of a basic polyhedron <math>P</math> where <math>n</math> is the number of vertices and <math>i</math> is the index of <math>P</math> in some fixed list of basic polyhedra with <math>n</math> vertices, followed by the symbols for the tangles <math>t_1</math> through <math>t_n</math> separated by dots.

For example, the knot [[4_1]] is denoted by "2 2", the knot [[9_5]] by "5 1 3", the link [[L5a1]] is denoted by "2 1 2", the link [[L9a24]] by "3 1,3,2" (all of them contain spaces between tangles), etc. A sequence of k pluses at the end of Conway symbol is denoted by +k, and the sequence of k minuses by +-k (e.g., knot [[10_76]] given in Conway notation as 3,3,2++ is denoted by "3,3,2+2", and the mirror of the link [[L9n21]] whose Conway notation is 3,2,2,2-- is given by "3,2,2,2+-2"). The space is used in the same way in all other symbols.

For the basic polyhedra with <math>N<10</math> crossings the standard notation is used (.1 , 6*, 8*, 9*, where the symbol for 6* can be ommitted). For example, the knot [[10_95]] is denoted by ".2 1 0.2.2", and [[10_101]] by "2 1..2..2". For higher values of <math>N</math> it is used notation in which the first number is the number of crossings, and the next is the ordering number of polyhedron (e.g., 101*, 102*, 103* for <math>N=10</math> denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for <math>N=11</math> denoting 11*, 11**, 11***, respectively, ''etc''.).

The order of basic polyhedra for <math>N=12</math> corresponds to the list in {{ref|Caudron}}, so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For <math>N>12</math> the database of basic polyhedra is produced from the list of simple 4-regular 4-edge-connected but not 3-connected plane graphs generated by Brendan McKay using the program "plantri" written by Gunnar Brinkmann and Brendan McKay (http://cs.anu.edu.au/~bdm/plantri/). PolyBase.m is automatically downloaded and it cointains basic polyhedra up to 16 crossings. In order to work with the basic polyhedra up tp 20 vertices, one needs to open an additional database PolyBaseN.m, for <math>N=17</math> to <math>N=20</math> (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for <math>N=17</math>).

Note: Together with the classical notation, Conway symbols are given in the book ''Knots and Links'' by D.~Rolfsen. However if you try to draw some knots or links from their Conway symbols the obtained projection might be non-isomorphic with the one given in Rolfsen, for example knot [[9_15]] denoted in Conway notation as 2 3 2 2 gives projection with 5, and not 4 digons.

{{note|Caudron}} A. Caudron, ''Classification des noeuds et des enlancements''. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982.

{{note|Conway}} J. H. Conway, ''An Enumeration of Knots and Links, and Some of Their Algebraic Properties.'' In Computation Problems in Abstract Algebra (Ed. J. Leech). Oxford, England: Pergamon Press, pp. 329-358, 1967.