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{{Link Page|
{{Link Page|
n = 8 |
n = 8 |
t = a |
t = <nowiki>a</nowiki> |
k = 12 |
k = 12 |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-7,3,-5,4,-6,2,-8:7,-1,8,-3,5,-4,6,-2/goTop.html |
KnotilusURL = http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-7,3,-5,4,-6,2,-8:7,-1,8,-3,5,-4,6,-2/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[8, Alternating, 12]]</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>8</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[8, Alternating, 12]]]</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[8, Alternating, 12]]</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[8, Alternating, 12]]</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[10, 1, 11, 2], X[16, 7, 9, 8], X[12, 3, 13, 4], X[14, 5, 15, 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>8</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[8, Alternating, 12]]]</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[8, Alternating, 12]]</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[10, 1, 11, 2], X[16, 7, 9, 8], X[12, 3, 13, 4], X[14, 5, 15, 6],
X[4, 13, 5, 14], X[6, 15, 7, 16], X[2, 9, 3, 10], X[8, 11, 1, 12]]</nowiki></pre></td></tr>
X[4, 13, 5, 14], X[6, 15, 7, 16], X[2, 9, 3, 10], X[8, 11, 1, 12]]</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[8, Alternating, 12]]</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, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[5]:=</code></td>
<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[8, Alternating, 12]]]</nowiki></pre></td></tr><tr><td></td><td align=left>[[Image:L8a12_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>In[7]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>KnotSignature[Link[8, Alternating, 12]]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[Link[8, Alternating, 12]]</nowiki></code></td></tr>
<tr align=left>
<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>-5</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[8, Alternating, 12]][q]</nowiki></pre></td></tr>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">Out[5]:=</code></td>
<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> -(21/2) -(19/2) 2 3 3 2 2 -(7/2)
<td><code style="white-space: pre; color: black; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>GaussCode[{1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}]</nowiki></code></td></tr>
</table>
<table><tr align=left>
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[6]:=</code></td>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>Show[DrawMorseLink[Link[8, Alternating, 12]]]</nowiki></code></td></tr>
<tr align=left><td></td><td>[[Image:L8a12_ML.gif]]</td></tr><tr align=left>
<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[8, Alternating, 12]]</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>-5</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[8, Alternating, 12]][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> -(21/2) -(19/2) 2 3 3 2 2 -(7/2)
-q + q - ----- + ----- - ----- + ----- - ---- + q -
-q + q - ----- + ----- - ----- + ----- - ---- + q -
17/2 15/2 13/2 11/2 9/2
17/2 15/2 13/2 11/2 9/2
Line 62: Line 98:
-(5/2)
-(5/2)
q</nowiki></pre></td></tr>
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[8, Alternating, 12]][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> -32 -30 -28 -26 -22 -18 -16 -12 -8
<td width=70px><code style="color: blue; border: 0px; padding: 0em">In[9]:=</code></td>
q + q + q + q + q + q + q + q + q</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[8, Alternating, 12]][a, z]</nowiki></pre></td></tr>
<td><code style="white-space: pre; color: red; border: 0px; padding: 0em; background-color: rgb(255,255,255);"><nowiki>A2Invariant[Link[8, Alternating, 12]][q]</nowiki></code></td></tr>
<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> 7 9
<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> -32 -30 -28 -26 -22 -18 -16 -12 -8
q + q + q + q + q + q + q + q + q</nowiki></code></td></tr>
</table>
<table><tr align=left>
<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[8, Alternating, 12]][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> 7 9
a a 5 7 9 5 3 7 3 9 3
a a 5 7 9 5 3 7 3 9 3
-(--) + -- - 3 a z - 4 a z + 3 a z - 4 a z - 4 a z + a z -
-(--) + -- - 3 a z - 4 a z + 3 a z - 4 a z - 4 a z + a z -
Line 73: Line 119:
5 5 7 5
5 5 7 5
a z - a z</nowiki></pre></td></tr>
a z - a z</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[8, Alternating, 12]][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> 7 9
<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[8, Alternating, 12]][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> 7 9
8 a a 5 7 9 11 13 6 2
8 a a 5 7 9 11 13 6 2
a - -- - -- - 3 a z + 5 a z + 5 a z - a z + 2 a z - a z -
a - -- - -- - 3 a z + 5 a z + 5 a z - a z + 2 a z - a z -
Line 87: Line 138:
9 5 11 5 6 6 8 6 10 6 7 7 9 7
9 5 11 5 6 6 8 6 10 6 7 7 9 7
4 a z - a z - a z - 2 a z - a z - a z - a z</nowiki></pre></td></tr>
4 a z - a z - a z - 2 a z - 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[8, Alternating, 12]][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> -6 -4 1 1 1 1 1 2
<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[8, Alternating, 12]][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> -6 -4 1 1 1 1 1 2
q + q + ------ + ------ + ------ + ------ + ------ + ------ +
q + q + ------ + ------ + ------ + ------ + ------ + ------ +
22 8 20 8 20 7 18 6 16 6 16 5
22 8 20 8 20 7 18 6 16 6 16 5
Line 97: Line 153:
------ + ------ + ------ + ------ + ------ + ------ + ----- + ----
------ + ------ + ------ + ------ + ------ + ------ + ----- + ----
14 5 14 4 12 4 12 3 10 3 10 2 8 2 6
14 5 14 4 12 4 12 3 10 3 10 2 8 2 6
q t q t q t q t q t q t q t q t</nowiki></pre></td></tr>
q t q t q t q t q t q t q t q t</nowiki></code></td></tr>
</table> }}
</table> }}

Revision as of 17:40, 1 September 2005

L8a11.gif

L8a11

L8a13.gif

L8a13

L8a12.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a12 at Knotilus!

L8a12 is in the Rolfsen table of links.


Link Presentations

[edit Notes on L8a12's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,7,9,8 X12,3,13,4 X14,5,15,6 X4,13,5,14 X6,15,7,16 X2,9,3,10 X8,11,1,12
Gauss code {1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}
A Braid Representative {{{braid_table}}}
A Morse Link Presentation L8a12 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in , , , ...) (db)
Jones polynomial (db)
Signature -5 (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 \
-8-7-6-5-4-3-2-10χ
-4        11
-6       110
-8      1  1
-10     11  0
-12    21   1
-14   11    0
-16  12     -1
-18  1      1
-2011       0
-221        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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L8a11

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L8a13