L11a427: Difference between revisions

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{{Link Page|
{{Link Page|
n = 11 |
n = 11 |
t = a |
t = <nowiki>a</nowiki> |
k = 427 |
k = 427 |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-11:3,-1,4,-5,10,-7:6,-2,7,-3,11,-6,8,-9,5,-4,9,-8/goTop.html |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-11:3,-1,4,-5,10,-7:6,-2,7,-3,11,-6,8,-9,5,-4,9,-8/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, Alternating, 427]]</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, Alternating, 427]]]</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>3</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, Alternating, 427]]</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, Alternating, 427]]</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[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[20, 7, 21, 8],
<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, Alternating, 427]]]</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>3</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, Alternating, 427]]</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[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[20, 7, 21, 8],
X[8, 19, 9, 20], X[16, 12, 17, 11], X[10, 13, 5, 14],
X[8, 19, 9, 20], X[16, 12, 17, 11], X[10, 13, 5, 14],
X[22, 18, 11, 17], X[18, 22, 19, 21], X[2, 9, 3, 10], X[4, 16, 1, 15]]</nowiki></pre></td></tr>
X[22, 18, 11, 17], X[18, 22, 19, 21], X[2, 9, 3, 10], X[4, 16, 1, 15]]</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, Alternating, 427]]</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, 2, -11}, {3, -1, 4, -5, 10, -7},
<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, Alternating, 427]]</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, 2, -11}, {3, -1, 4, -5, 10, -7},
{6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}]</nowiki></pre></td></tr>
{6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}]</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, Alternating, 427]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:L11a427_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, Alternating, 427]]</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>-2</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, Alternating, 427]][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, Alternating, 427]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:L11a427_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> -7 3 7 10 16 16 17 2 3 4
<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, Alternating, 427]]</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>-2</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, Alternating, 427]][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> -7 3 7 10 16 16 17 2 3 4
-15 + q - -- + -- - -- + -- - -- + -- + 11 q - 7 q + 4 q - q
-15 + q - -- + -- - -- + -- - -- + -- + 11 q - 7 q + 4 q - q
6 5 4 3 2 q
6 5 4 3 2 q
q q q q q</nowiki></pre></td></tr>
q q q 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, Alternating, 427]][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> -22 5 2 4 7 2 6 2 4 10 12
<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, Alternating, 427]][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> -22 5 2 4 7 2 6 2 4 10 12
q + --- + --- + --- + --- + -- + -- - 4 q + 3 q + 2 q - q
q + --- + --- + --- + --- + -- + -- - 4 q + 3 q + 2 q - q
16 14 12 10 8 6
16 14 12 10 8 6
q q q q q q</nowiki></pre></td></tr>
q q q q 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, Alternating, 427]][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> 2 4 6 2
<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, Alternating, 427]][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> 2 4 6 2
-2 2 4 6 a 2 a a z 2 2 4 2
-2 2 4 6 a 2 a a z 2 2 4 2
-2 + a + 4 a - 4 a + a + -- - ---- + -- - -- + 3 a z - 4 a z +
-2 + a + 4 a - 4 a + a + -- - ---- + -- - -- + 3 a z - 4 a z +
Line 83: Line 129:
a z + 2 z - -- + 2 a z - 2 a z + z + a z
a z + 2 z - -- + 2 a z - 2 a z + z + a z
2
2
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, Alternating, 427]][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> 2 4 6 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, Alternating, 427]][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> 2 4 6 3 5
-2 2 4 6 a 2 a a 2 a 2 a 2 z
-2 2 4 6 a 2 a a 2 a 2 a 2 z
-1 - a - 3 a - 6 a - 4 a + -- + ---- + -- - ---- - ---- - --- -
-1 - a - 3 a - 6 a - 4 a + -- + ---- + -- - ---- - ---- - --- -
Line 124: Line 175:
4 8 5 z 9 3 9 10 2 10
4 8 5 z 9 3 9 10 2 10
7 a z + ---- + 10 a z + 5 a z + 2 z + 2 a z
7 a z + ---- + 10 a z + 5 a z + 2 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[12]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Kh[Link[11, Alternating, 427]][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>10 9 1 1 3 4 3 6 4
<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, Alternating, 427]][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>10 9 1 1 3 4 3 6 4
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- +
-- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- +
3 q 15 6 13 6 13 5 11 4 9 4 9 3 7 3
3 q 15 6 13 6 13 5 11 4 9 4 9 3 7 3
Line 137: Line 193:
3 3 5 3 5 4 7 4 9 5
3 3 5 3 5 4 7 4 9 5
3 q t + 4 q t + q t + 3 q t + q t</nowiki></pre></td></tr>
3 q t + 4 q t + q t + 3 q t + q t</nowiki></code></td></tr>
</table> }}
</table> }}

Revision as of 18:48, 1 September 2005

L11a426.gif

L11a426

L11a428.gif

L11a428

L11a427.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a427 at Knotilus!


Link Presentations

[edit Notes on L11a427's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,5,15,6 X20,7,21,8 X8,19,9,20 X16,12,17,11 X10,13,5,14 X22,18,11,17 X18,22,19,21 X2,9,3,10 X4,16,1,15
Gauss code {1, -10, 2, -11}, {3, -1, 4, -5, 10, -7}, {6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}
A Braid Representative {{{braid_table}}}
A Morse Link Presentation L11a427 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in , , , ...) (db)
Jones polynomial (db)
Signature -2 (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
9           1-1
7          3 3
5         41 -3
3        73  4
1       84   -4
-1      97    2
-3     910     1
-5    77      0
-7   410       6
-9  36        -3
-11  4         4
-1313          -2
-151           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.

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