L11a353: Difference between revisions

From Knot Atlas
Jump to navigationJump to search
No edit summary
 
No edit summary
Line 1: Line 1:
<!-- WARNING! WARNING! WARNING!
<!-- This page was generated from the splice template "Link_Splice_Template". Please do not edit! -->
<!-- This page was generated from the splice template [[Link_Splice_Base]]. Please do not edit!
<!-- --> <!--
<!-- You probably want to edit the template referred to immediately below. (See [[Category:Knot Page Template]].)
-->
<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Link_Splice_Base]]. -->
<!-- <math>\text{Null}</math> -->
<!-- <math>\text{Null}</math> -->
<!-- WARNING! WARNING! WARNING!
<!-- This page was generated from the splice template [[Link Splice Template]]. Please do not edit!
<!-- Almost certainly, you want to edit [[Template:Link Page]], which actually produces this page.
<!-- The text below simply calls [[Template:Link Page]] setting the values of all the parameters appropriately.
<!-- This page itself was created by running [[Media:KnotPageSpliceRobot.nb]] on [[Link Splice Template]]. -->
<!-- <math>\text{Null}</math> -->
{{Link Page|
{{Link Page|
n = 11 |
n = 11 |
Line 35: Line 44:
<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 29, 2005, 15:33:11)...</td></tr>
<tr valign=top><td colspan=2>Loading KnotTheory` (version of August 28, 2005, 22:58:49)...</td></tr>
<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, Alternating, 353]]</nowiki></pre></td></tr>
<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, Alternating, 353]]</nowiki></pre></td></tr>
<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>Out[2]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>11</nowiki></pre></td></tr>
Line 50: Line 59:
{10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}]</nowiki></pre></td></tr>
{10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}]</nowiki></pre></td></tr>
<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>BR[Link[11, Alternating, 353]]</nowiki></pre></td></tr>
<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, Alternating, 353]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:L11a353_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>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[6]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki>BR[Link[11, Alternating, 353]]</nowiki></pre></td></tr>
<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, Alternating, 353]]</nowiki></pre></td></tr>
<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>Show[DrawMorseLink[Link[11, Alternating, 353]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:L11a353_ML.gif]]</td></tr><tr valign=top><td><tt><font color=blue>Out[7]=</font></tt><td><tt><font color=black>-Graphics-</font></tt></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>
<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>alex = Alexander[Link[11, Alternating, 353]][t]</nowiki></pre></td></tr>
<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, Alternating, 353]][q]</nowiki></pre></td></tr>
<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>ComplexInfinity</nowiki></pre></td></tr>
<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) 4 9 15 21 24
<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>Conway[Link[11, Alternating, 353]][z]</nowiki></pre></td></tr>
<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>ComplexInfinity</nowiki></pre></td></tr>
<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>Select[AllKnots[], (alex === Alexander[#][t])&]</nowiki></pre></td></tr>
<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>{}</nowiki></pre></td></tr>
<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>{KnotDet[Link[11, Alternating, 353]], KnotSignature[Link[11, Alternating, 353]]}</nowiki></pre></td></tr>
<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>{Infinity, 1}</nowiki></pre></td></tr>
<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>J=Jones[Link[11, Alternating, 353]][q]</nowiki></pre></td></tr>
<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> -(11/2) 4 9 15 21 24
-q + ---- - ---- + ---- - ---- + ------- - 25 Sqrt[q] +
-q + ---- - ---- + ---- - ---- + ------- - 25 Sqrt[q] +
9/2 7/2 5/2 3/2 Sqrt[q]
9/2 7/2 5/2 3/2 Sqrt[q]
Line 69: Line 70:
3/2 5/2 7/2 9/2 11/2
3/2 5/2 7/2 9/2 11/2
21 q - 17 q + 10 q - 4 q + q</nowiki></pre></td></tr>
21 q - 17 q + 10 q - 4 q + q</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[13]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>A2Invariant[Link[11, Alternating, 353]][q]</nowiki></pre></td></tr>
<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, Alternating, 353]][q]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[13]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> -18 3 2 2 6 2 2 2 4 6 8
<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 3 2 2 6 2 2 2 4 6 8
1 + q - --- + --- - -- + -- - -- + -- - 2 q + 6 q - 3 q + 5 q +
1 + q - --- + --- - -- + -- - -- + -- - 2 q + 6 q - 3 q + 5 q +
14 12 8 6 4 2
14 12 8 6 4 2
Line 77: Line 78:
10 12 14 16
10 12 14 16
q - 4 q + 2 q - q</nowiki></pre></td></tr>
q - 4 q + 2 q - q</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[14]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>HOMFLYPT[Link[11, Alternating, 353]][a, z]</nowiki></pre></td></tr>
<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, Alternating, 353]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[14]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3 3
<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 3
1 a 3 z 7 z 3 5 2 z 9 z 3
1 a 3 z 7 z 3 5 2 z 9 z 3
-(---) + - + --- - --- + 6 a z - 3 a z + a z + ---- - ---- + 7 a z -
-(---) + - + --- - --- + 6 a z - 3 a z + a z + ---- - ---- + 7 a z -
Line 89: Line 90:
3 a a
3 a a
a</nowiki></pre></td></tr>
a</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[15]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kauffman[Link[11, Alternating, 353]][a, z]</nowiki></pre></td></tr>
<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, Alternating, 353]][a, z]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[15]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 2
<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> 2 2
1 a 5 z 10 z 3 5 2 3 z z
1 a 5 z 10 z 3 5 2 3 z z
1 - --- - - + --- + ---- + 6 a z + 2 a z + a z + 10 z - ---- + -- +
1 - --- - - + --- + ---- + 6 a z + 2 a z + a z + 10 z - ---- + -- +
Line 128: Line 129:
10 2 10
10 2 10
3 z - 3 a z</nowiki></pre></td></tr>
3 z - 3 a z</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[16]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>{Vassiliev[2][Link[11, Alternating, 353]], Vassiliev[3][Link[11, Alternating, 353]]}</nowiki></pre></td></tr>
<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, Alternating, 353]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[16]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 3
<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 1 3 1 6 3 9 6
{0, -(-)}
2</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>In[17]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Link[11, Alternating, 353]][q, t]</nowiki></pre></td></tr>
<tr valign=top><td><pre style="color: blue; border: 0px; padding: 0em"><nowiki>Out[17]=&nbsp;&nbsp;</nowiki></pre></td><td><pre style="color: black; border: 0px; padding: 0em"><nowiki> 2 1 3 1 6 3 9 6
13 + 14 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- +
13 + 14 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- +
12 6 10 5 8 5 8 4 6 4 6 3 4 3
12 6 10 5 8 5 8 4 6 4 6 3 4 3

Revision as of 12:05, 31 August 2005

L11a352.gif

L11a352

L11a354.gif

L11a354

L11a353.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a353 at Knotilus!


Link Presentations

[edit Notes on L11a353's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X16,6,17,5 X22,8,11,7 X20,15,21,16 X14,21,15,22 X6,14,7,13 X4,20,5,19 X18,9,19,10 X2,11,3,12 X10,17,1,18
Gauss code {1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}
A Braid Representative {{{braid_table}}}
A Morse Link Presentation L11a353 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 \
-6-5-4-3-2-1012345χ
12           1-1
10          3 3
8         71 -6
6        103  7
4       117   -4
2      1410    4
0     1213     1
-2    912      -3
-4   612       6
-6  39        -6
-8 16         5
-10 3          -3
-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).

Back to the top.

L11a352.gif

L11a352

L11a354.gif

L11a354