L11n243: Difference between revisions

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{{Link Page|
{{Link Page|
n = 11 |
n = 11 |
t = n |
t = <nowiki>n</nowiki> |
k = 243 |
k = 243 |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,-4,5,-3,4,2,-11,-5,3:10,-1,6,-9,11,-2,7,-8,9,-6,8,-7/goTop.html |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,-4,5,-3,4,2,-11,-5,3:10,-1,6,-9,11,-2,7,-8,9,-6,8,-7/goTop.html |
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<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
</tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 28, 2005, 22:58:49)...</td></tr>
<tr valign=top><td colspan=2><nowiki>Loading KnotTheory` (version of August 29, 2005, 15:33:11)...</nowiki></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[2]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Crossings[Link[11, NonAlternating, 243]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>11</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Length[Skeleton[Link[11, NonAlternating, 243]]]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[2]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[3]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>2</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Crossings[Link[11, NonAlternating, 243]]</nowiki></code></td></tr>
<tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[4]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Link[11, NonAlternating, 243]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[4]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[5, 1, 6, 10], X[3, 7, 4, 6],
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[2]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>11</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[3]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Length[Skeleton[Link[11, NonAlternating, 243]]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[3]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>2</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[4]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[Link[11, NonAlternating, 243]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[4]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[5, 1, 6, 10], X[3, 7, 4, 6],
X[9, 5, 10, 4], X[20, 14, 21, 13], X[22, 17, 11, 18],
X[9, 5, 10, 4], X[20, 14, 21, 13], X[22, 17, 11, 18],
Line 54: Line 69:
X[18, 21, 19, 22], X[14, 20, 15, 19], X[2, 11, 3, 12],
X[18, 21, 19, 22], X[14, 20, 15, 19], X[2, 11, 3, 12],
X[8, 15, 9, 16]]</nowiki></pre></td></tr>
X[8, 15, 9, 16]]</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[5]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>GaussCode[Link[11, NonAlternating, 243]]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[5]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>GaussCode[{1, -10, -4, 5, -3, 4, 2, -11, -5, 3},
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Link[11, NonAlternating, 243]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[{1, -10, -4, 5, -3, 4, 2, -11, -5, 3},
{10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}]</nowiki></pre></td></tr>
{10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}]</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[6]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Show[DrawMorseLink[Link[11, NonAlternating, 243]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:L11n243_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[6]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotSignature[Link[11, NonAlternating, 243]]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[7]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>1</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[8]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>J=Jones[Link[11, NonAlternating, 243]][q]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Link[11, NonAlternating, 243]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:L11n243_ML.gif]]</td></tr><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[8]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -(11/2) 2 5 6 7 7 3/2
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[6]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>-Graphics-</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[7]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>KnotSignature[Link[11, NonAlternating, 243]]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[7]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>1</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[8]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>J=Jones[Link[11, NonAlternating, 243]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[8]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -(11/2) 2 5 6 7 7 3/2
-q + ---- - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 5 q -
-q + ---- - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 5 q -
9/2 7/2 5/2 3/2 Sqrt[q]
9/2 7/2 5/2 3/2 Sqrt[q]
Line 69: Line 105:
5/2 7/2
5/2 7/2
3 q + q</nowiki></pre></td></tr>
3 q + q</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[9]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Link[11, NonAlternating, 243]][q]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[9]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -18 2 -14 3 -8 2 2 2 4 6 8
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Link[11, NonAlternating, 243]][q]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[9]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -18 2 -14 3 -8 2 2 2 4 6 8
1 + q + --- + q + --- - q - -- + -- + 2 q + 2 q - q + q -
1 + q + --- + q + --- - q - -- + -- + 2 q + 2 q - q + q -
16 12 4 2
16 12 4 2
Line 77: Line 118:
10 12
10 12
q - q</nowiki></pre></td></tr>
q - q</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[10]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Link[11, NonAlternating, 243]][a, z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[10]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 5
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[10]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>HOMFLYPT[Link[11, NonAlternating, 243]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[10]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 5
1 4 6 a 5 a 2 a z 7 z 3 5
1 4 6 a 5 a 2 a z 7 z 3 5
---- - --- + --- - ---- + ---- + -- - --- + 11 a z - 8 a z + a z -
---- - --- + --- - ---- + ---- + -- - --- + 11 a z - 8 a z + a z -
Line 88: Line 134:
3 z 3 3 3 5
3 z 3 3 3 5
---- + 8 a z - 3 a z + 2 a z
---- + 8 a z - 3 a z + 2 a z
a</nowiki></pre></td></tr>
a</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Link[11, NonAlternating, 243]][a, z]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[11]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 5
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[11]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kauffman[Link[11, NonAlternating, 243]][a, z]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[11]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> 3 5
-4 3 2 4 1 4 6 a 5 a 2 a 3 z 18 z
-4 3 2 4 1 4 6 a 5 a 2 a 3 z 18 z
3 + a + -- + a + a - ---- - --- - --- - ---- - ---- + --- + ---- +
3 + a + -- + a + a - ---- - --- - --- - ---- - ---- + --- + ---- +
Line 120: Line 171:
2 8 4 8 9 3 9
2 8 4 8 9 3 9
6 a z - 2 a z - a z - a z</nowiki></pre></td></tr>
6 a z - 2 a z - a z - a z</nowiki></code></td></tr>
</table>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Link[11, NonAlternating, 243]][q, t]</nowiki></pre></td></tr>
<table><tr align=left>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[12]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -2 2 1 1 1 4 2 3
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[12]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Kh[Link[11, NonAlternating, 243]][q, t]</nowiki></code></td></tr>
<tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[12]:=</code></td>
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki> -2 2 1 1 1 4 2 3
4 + q + 4 q + ------ + ------ + ----- + ----- + ----- + ----- +
4 + q + 4 q + ------ + ------ + ----- + ----- + ----- + ----- +
12 6 10 5 8 5 8 4 6 4 6 3
12 6 10 5 8 5 8 4 6 4 6 3
Line 133: Line 189:
6 2 8 3
6 2 8 3
2 q t + q t</nowiki></pre></td></tr>
2 q t + q t</nowiki></code></td></tr>
</table> }}
</table> }}

Revision as of 17:42, 1 September 2005

L11n242.gif

L11n242

L11n244.gif

L11n244

L11n243.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n243 at Knotilus!


Link Presentations

[edit Notes on L11n243's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X20,14,21,13 X22,17,11,18 X18,21,19,22 X14,20,15,19 X2,11,3,12 X8,15,9,16
Gauss code {1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, 6, -9, 11, -2, 7, -8, 9, -6, 8, -7}
A Braid Representative {{{braid_table}}}
A Morse Link Presentation L11n243 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in , , , ...) (db)
Jones polynomial (db)
Signature 1 (db)
HOMFLY-PT polynomial (db)
Kauffman polynomial (db)

Khovanov Homology

The coefficients of the monomials are shown, along with their alternating sums (fixed , alternation over ).   
\ r
  \  
j \
-6-5-4-3-2-10123χ
8         1-1
6        2 2
4       31 -2
2      42  2
0     44   0
-2    341   0
-4   34     1
-6  23      -1
-8 14       3
-10 1        -1
-121         1
Integral Khovanov Homology

(db, data source)

  

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L11n242

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