Knot Atlas - User contributions [en]

Conway Notation

2006-03-13T06:30:05Z

Radmila: /* Some generalities about the Conway notation */

{{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). See [[Using the LinKnot package]] for more information.

{{Startup Note}}

As in the section [[Using the LinKnot package]], the first step is to add ''LinKnot'' to the Mathematica search path. This path will likely be different on your computer.



{{In|
n = 2 |
in = <nowiki>AppendTo[$Path, "C:/bin/LinKnot/"];</nowiki>}}




{{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.

File:Tangle-1.jpg

2006-03-13T06:17:58Z

Radmila: Elementary tangle -1

Elementary tangle -1

File:Tangle1.jpg

2006-03-13T06:17:34Z

Radmila: Elementary tangle 1

Elementary tangle 1

File:Tangle0.jpg

2006-03-13T06:17:01Z

Radmila: Elementary tangle 0

Elementary tangle 0

Conway Notation

2006-03-10T01:20:47Z

Radmila:

{{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). See [[Using the LinKnot package]] for more information.

{{Startup Note}}

As in the section [[Using the LinKnot package]], the first step is to add ''LinKnot'' to the Mathematica search path. This path will likely be different on your computer.



{{In|
n = 2 |
in = <nowiki>AppendTo[$Path, "C:/bin/LinKnot/"];</nowiki>}}




{{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 <math>0</math>, <math>1</math> and <math>-1</math>.

Pic

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.

File:ConwayRamification.jpg

2006-03-10T01:11:15Z

Radmila: Conway notation-ramification

Conway notation-ramification

File:033.jpg

2006-03-10T01:09:51Z

Radmila: Conway notation-sum and product

Conway notation-sum and product

Conway Notation

2006-03-09T16:23:42Z

Radmila:

{{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). See [[Using the LinKnot package]] for more information.

{{Startup Note}}

As in the section [[Using the LinKnot package]], the first step is to add ''LinKnot'' to the Mathematica search path. This path will likely be different on your computer.



{{In|
n = 2 |
in = <nowiki>AppendTo[$Path, "C:/bin/LinKnot/"];</nowiki>}}




{{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 <math>0</math>, <math>1</math> and <math>-1</math>.

Pic

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>.

Pic

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.

Conway Notation

2006-02-18T17:23:36Z

Radmila: /* Some generalities about the Conway notation */

{{Manual TOC Sidebar}}

====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 <math>0</math>, <math>1</math> and <math>-1</math>.

Pic

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>.

Pic

A ''rational tangle'' is any tangle obtained from the elementary tangles using the above operations. A ''rational knot'' or a ''rational link'' is the numerator the 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", knot [[9_5]] by "5 1 3", the link [[L5a1]] is denoted by "2 1 2" , 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 link 9_17^3 given in Conway notation as 3,2,2,2-- by "3,2,2,2+-2"). The space is used in the same way in all other symbols. For example, the knot 10_133 is denoted by "2 3,2 1,2+-1", and the knot 10_154 by "(2 1,2) -(2 1,2)" (with spaces).

For the basic polyhedra with N<10 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 N 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 N=10 denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for N=11 denoting 11*, 11**, 11***, respectively, etc.).

The order of basic polyhedra for N=12 corresponds to their list made by A.Caudron (Caudron A.: Classification des noeuds et des enlancements. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982. ), so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For N>12 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 N=17 to N=20 (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for N=17).

Note: Together with the classical notation, Conway symbols are given in the book {\it 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 $9_{15}$ gives projection with 5 not 4 digons.

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

Converting Conway notation to other forms known to KnotTheory` (which is a necessary first step in using most of the KnotTheory` functions on knots in Conway notation) requires the package LinKnots` (by M.Ochiai and N.Imafuji, S. Jablan and R. Sazdanovic). See [[Using the package LinKnots`]] for more information.

{{Startup Note}}



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




{{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>}}


{{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.

Conway Notation

2006-02-18T16:44:31Z

Radmila:

{{Manual TOC Sidebar}}

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

Conway notation was introduced by J.H. Conway in 1967 (see {{ref|Conway}}).
...
The main building block is tangle. A {\it tangle} in a knot or link projection is a region in the projectionplane $\Re ^2$ (or on the sphere $S^3$) surrounded with a circle such that the projection intersects with the circle exactly four times.The elementary tangles are 0, 1 and $-1$.

Pic

Tangles could be combined and modified by two operations: {\it sum} and {\it product},leading from tangles $a$, $b$ to the new tangles $a+b$, $-a$, $a\,b$, where $-a$ is the image of $a$ in NW-SE mirror line, $a\,b = -a+b$, and $-a = a\,0$. A third operation: {\it ramification} is defined as $a,b = -a-b$.

Pic

A {\it rational tangle} is any tangle obtained from elementary tangles using described operations.
A {\it rational knot or link} is a numerator closure of a rational tangle.
A knot or link is called {\it algebraic} if it can be obtained as a closure of a tangle obtained from rational tangles using operations product and sum. Knot or links that can not be obtained in this way are called {\it non-algebraic}. Conway notation for non-algebraic knots and links is a bit more complicated. It consists of symbol of basic polyhedron $P^*$ where $P$=$ni$ where $n$ is the number of vertices and $i$ is the index in the list of basic polyhedra with $n$ crossings.A 4-valent graph without digons is called a {\it basic polyhedron}or more precisey it is 4-regular 4-edge-connected, at least 2-vertex connected plane graph.
Non-algebraic knots and links can be obtained by substituting tangles in $P^*$ by substituting tangles $t_1$, $\ldots$, $t_k$ in appropriate places is denoted by $P^*t_1\ldots t_k$, where the number of dots between two successive tangles shows the number of omitted substituents of value 1.

For example, the knot 4_1 is denoted by "2 2", knot 9_5 by "5 1 3", link 5_1^2 is denoted by "2 1 2" , link 9_21^2 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 link 9_17^3 given in Conway notation as 3,2,2,2-- by "3,2,2,2+-2"). The space is used in the same way in all other symbols. For example, the knot 10_133 is denoted by "2 3,2 1,2+-1", and the knot 10_154 by "(2 1,2) -(2 1,2)" (with spaces).

For the basic polyhedra with N<10 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 N 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 N=10 denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for N=11 denoting 11*, 11**, 11***, respectively, etc.).

The order of basic polyhedra for N=12 corresponds to their list made by A.Caudron (Caudron A.: Classification des noeuds et des enlancements. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982. ), so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For N>12 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 N=17 to N=20 (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for N=17).

Note: Together with the classical notation, Conway symbols are given in the book {\it 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 $9_{15}$ gives projection with 5 not 4 digons.

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

Converting Conway notation to other forms known to KnotTheory` (which is a necessary first step in using most of the KnotTheory` functions on knots in Conway notation) requires the package LinKnots` (by M.Ochiai and N.Imafuji, S. Jablan and R. Sazdanovic). See [[Using the package LinKnots`]] for more information.

{{Startup Note}}



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




{{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>}}


{{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.

Conway Notation

2006-02-18T16:35:34Z

Radmila:

{{Manual TOC Sidebar}}

Converting Conway notation to other forms known to KnotTheory` (which is a necessary first step in using most of the KnotTheory` functions on knots in Conway notation) requires the package LinKnots` (by M.Ochiai and N.Imafuji, S. Jablan and R. Sazdanovic). See [[Using the package LinKnots`]] for more information.

{{Startup Note}}



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




{{HelpAndAbout|
n = 2 |
n1 = 3 |
in = <nowiki>Kauffman</nowiki> |
out= <nowiki>Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.</nowiki> |
about= <nowiki>The Kauffman program was written by Scott Morrison.</nowiki>}}


(* to be updated*)

Conway notation was introduced by J.H.~Conway in 1967 (Conway, 1970).
...
The main building block is tangle. A {\it tangle} in a knot or link projection is a region in the projectionplane $\Re ^2$ (or on the sphere $S^3$) surrounded with a circle such that the projection intersects with the circle exactly four times.The elementary tangles are 0, 1 and $-1$.

Pic

Tangles could be combined and modified by two operations: {\it sum} and {\it product},leading from tangles $a$, $b$ to the new tangles $a+b$, $-a$, $a\,b$, where $-a$ is the image of $a$ in NW-SE mirror line, $a\,b = -a+b$, and $-a = a\,0$. A third operation: {\it ramification} is defined as $a,b = -a-b$.

Pic

A {\it rational tangle} is any tangle obtained from elementary tangles using described operations.
A {\it rational knot or link} is a numerator closure of a rational tangle.
A knot or link is called {\it algebraic} if it can be obtained as a closure of a tangle obtained from rational tangles using operations product and sum. Knot or links that can not be obtained in this way are called {\it non-algebraic}. Conway notation for non-algebraic knots and links is a bit more complicated. It consists of symbol of basic polyhedron $P^*$ where $P$=$ni$ where $n$ is the number of vertices and $i$ is the index in the list of basic polyhedra with $n$ crossings.A 4-valent graph without digons is called a {\it basic polyhedron}or more precisey it is 4-regular 4-edge-connected, at least 2-vertex connected plane graph.
Non-algebraic knots and links can be obtained by substituting tangles in $P^*$ by substituting tangles $t_1$, $\ldots$, $t_k$ in appropriate places is denoted by $P^*t_1\ldots t_k$, where the number of dots between two successive tangles shows the number of omitted substituents of value 1.

For example, the knot 4_1 is denoted by "2 2", knot 9_5 by "5 1 3", link 5_1^2 is denoted by "2 1 2" , link 9_21^2 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 link 9_17^3 given in Conway notation as 3,2,2,2-- by "3,2,2,2+-2"). The space is used in the same way in all other symbols. For example, the knot 10_133 is denoted by "2 3,2 1,2+-1", and the knot 10_154 by "(2 1,2) -(2 1,2)" (with spaces).

For the basic polyhedra with N<10 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 N 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 N=10 denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for N=11 denoting 11*, 11**, 11***, respectively, etc.).

The order of basic polyhedra for N=12 corresponds to their list made by A.Caudron (Caudron A.: Classification des noeuds et des enlancements. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982. ), so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For N>12 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 N=17 to N=20 (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for N=17).

Note: Together with the classical notation, Conway symbols are given in the book {\it 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 $9_{15}$ gives projection with 5 not 4 digons.

Conway Notation

2006-02-18T16:26:12Z

Radmila:

{{Manual TOC Sidebar}}

Converting Conway notation to other forms known to KnotTheory` (which is a necessary first step in using most of the KnotTheory` functions on knots in Conway notation) requires the package LinKnots` (by M.Ochiai and N.Imafuji, S. Jablan and R. Sazdanovic). See [[Using the package LinKnots`]] for more information.

{{Startup Note}}



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


(* to be updated*)

Conway notation was introduced by J.H.~Conway in 1967 (Conway, 1970).
...
The main building block is tangle. A {\it tangle} in a knot or link projection is a region in the projectionplane $\Re ^2$ (or on the sphere $S^3$) surrounded with a circle such that the projection intersects with the circle exactly four times.The elementary tangles are 0, 1 and $-1$.

Pic

Tangles could be combined and modified by two operations: {\it sum} and {\it product},leading from tangles $a$, $b$ to the new tangles $a+b$, $-a$, $a\,b$, where $-a$ is the image of $a$ in NW-SE mirror line, $a\,b = -a+b$, and $-a = a\,0$. A third operation: {\it ramification} is defined as $a,b = -a-b$.

Pic

A {\it rational tangle} is any tangle obtained from elementary tangles using described operations.
A {\it rational knot or link} is a numerator closure of a rational tangle.
A knot or link is called {\it algebraic} if it can be obtained as a closure of a tangle obtained from rational tangles using operations product and sum. Knot or links that can not be obtained in this way are called {\it non-algebraic}. Conway notation for non-algebraic knots and links is a bit more complicated. It consists of symbol of basic polyhedron $P^*$ where $P$=$ni$ where $n$ is the number of vertices and $i$ is the index in the list of basic polyhedra with $n$ crossings.A 4-valent graph without digons is called a {\it basic polyhedron}or more precisey it is 4-regular 4-edge-connected, at least 2-vertex connected plane graph.
Non-algebraic knots and links can be obtained by substituting tangles in $P^*$ by substituting tangles $t_1$, $\ldots$, $t_k$ in appropriate places is denoted by $P^*t_1\ldots t_k$, where the number of dots between two successive tangles shows the number of omitted substituents of value 1.

For example, the knot 4_1 is denoted by "2 2", knot 9_5 by "5 1 3", link 5_1^2 is denoted by "2 1 2" , link 9_21^2 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 link 9_17^3 given in Conway notation as 3,2,2,2-- by "3,2,2,2+-2"). The space is used in the same way in all other symbols. For example, the knot 10_133 is denoted by "2 3,2 1,2+-1", and the knot 10_154 by "(2 1,2) -(2 1,2)" (with spaces).

For the basic polyhedra with N<10 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 N 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 N=10 denoting 10*, 10**, 10***, respectively, and 111*, 112*, 113* for N=11 denoting 11*, 11**, 11***, respectively, etc.).

The order of basic polyhedra for N=12 corresponds to their list made by A.Caudron (Caudron A.: Classification des noeuds et des enlancements. Public. Math. d'Orsay 82. Orsay: Univ. Paris Sud, Dept. Math., 1982. ), so as 121* till 1212* are denoted the basic polyhedra originally titled as 12A-12L. For N>12 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 N=17 to N=20 (by writing, e.g. <<PolyBase17.m or Needs["PolyBase17.m"] for N=17).

Note: Together with the classical notation, Conway symbols are given in the book {\it 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 $9_{15}$ gives projection with 5 not 4 digons.

Conway Notation

2006-02-18T16:19:21Z

Radmila:

Using the LinKnot package

2006-02-18T15:58:04Z

Radmila:

{{Manual TOC Sidebar}}

The Mathematica package 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 notations, 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 [http://katlas.math.toronto.edu/svn/LinKnot/tags/stable/] (a full LinKnot manual page is in the zip file, in the mathematica notebook <tt>K2KL.nb</tt>). Install this wherever you like, and in your Mathematica session 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. (LinKnots` 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]].

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 LinKnotDirectory[].



{{HelpLine|
n = 3 |
in = <nowiki>LinKnotDirectory</nowiki> |
out= <math>\textrm{RemoveWhitespace}(\textrm{LinKnotDirectory}\textrm{::}\textrm{usage})</math>}}

Using the LinKnot package

2006-02-18T15:34:24Z

Radmila:

{{Manual TOC Sidebar}}

The Mathematica package 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 notations, 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 [http://katlas.math.toronto.edu/svn/LinKnot/tags/stable/] (a full LinKnot manual page is in the zip file, in the mathematica notebook <tt>K2KL.nb</tt>). Install this wherever you like, and in your Mathematica session issue a command like




After you've done this, everything should just work. (LinKnots` 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]].

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 LinKnotDirectory[].

User:Radmila

2006-02-18T15:25:14Z

Radmila:

I'm Radmila, find me at http://www.linknotess.com.