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{{Khovanov Homology|table=<table border=1>
<tr align=center>
<td width=12.5%><table cellpadding=0 cellspacing=0>
 <tr><td>\</td><td>&nbsp;</td><td>r</td></tr>
<tr><td>&nbsp;</td><td>&nbsp;\&nbsp;</td><td>&nbsp;</td></tr>
<tr><td>j</td><td>&nbsp;</td><td>\</td></tr>
</table></td>
 <td width=6.25%>-9</td  ><td width=6.25%>-8</td  ><td width=6.25%>-7</td  ><td width=6.25%>-6</td  ><td width=6.25%>-5</td  ><td width=6.25%>-4</td  ><td width=6.25%>-3</td  ><td width=6.25%>-2</td  ><td width=6.25%>-1</td  ><td width=6.25%>0</td  ><td width=6.25%>1</td  ><td width=6.25%>2</td  ><td width=12.5%>&chi;</td></tr>
<tr align=center><td>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td>1</td></tr>
<tr align=center><td>-1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-3</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>5</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>4</td></tr>
<tr align=center><td>-5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>7</td><td bgcolor=yellow>3</td><td>&nbsp;</td><td>&nbsp;</td><td>-4</td></tr>
<tr align=center><td>-7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>9</td><td bgcolor=yellow>4</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>5</td></tr>
<tr align=center><td>-9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>9</td><td bgcolor=yellow>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-2</td></tr>
<tr align=center><td>-11</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>9</td><td bgcolor=yellow>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>0</td></tr>
<tr align=center><td>-13</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>7</td><td bgcolor=yellow>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>-15</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>5</td><td bgcolor=yellow>9</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-4</td></tr>
<tr align=center><td>-17</td><td>&nbsp;</td><td>&nbsp;</td><td bgcolor=yellow>2</td><td bgcolor=yellow>7</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>5</td></tr>
<tr align=center><td>-19</td><td>&nbsp;</td><td bgcolor=yellow>1</td><td bgcolor=yellow>5</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-4</td></tr>
<tr align=center><td>-21</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>2</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>2</td></tr>
<tr align=center><td>-23</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
</table>}}

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<table>
<tr valign=top>
<td><pre style="color: blue; border: 0px; padding: 0em">In[1]:=&nbsp;&nbsp;&nbsp;&nbsp;</pre></td>
<td align=left><pre style="color: red; border: 0px; padding: 0em">&lt;&lt; KnotTheory`</pre></td>
</tr>
<tr valign=top><td colspan=2><pre style="border: 0px; padding: 0em">Loading KnotTheory` (version of August 17, 2005, 14:44:34)...</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[Knot[11, Alternating, 48]]</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>In[3]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>PD[Knot[11, Alternating, 48]]</nowiki></pre></td></tr>
<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>PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[10, 8, 11, 7], 
 
  X[2, 9, 3, 10], X[16, 11, 17, 12], X[20, 13, 21, 14], 
 
  X[6, 15, 7, 16], X[22, 17, 1, 18], X[12, 19, 13, 20], 
 
  X[18, 21, 19, 22]]</nowiki></pre></td></tr>
<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>GaussCode[Knot[11, Alternating, 48]]</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>GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -3, 8, -6, 9, 
 
  -11, 10, -7, 11, -9]</nowiki></pre></td></tr>
<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>BR[Knot[11, Alternating, 48]]</nowiki></pre></td></tr>
<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>BR[Knot[11, Alternating, 48]]</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>alex = Alexander[Knot[11, Alternating, 48]][t]</nowiki></pre></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>     2    12   26              2      3
33 - -- + -- - -- - 26 t + 12 t  - 2 t
      3    2   t
     t    t</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>Conway[Knot[11, Alternating, 48]][z]</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      6
1 + 4 z  - 2 z</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>Select[AllKnots[], (alex === Alexander[#][t])&]</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>{Knot[11, Alternating, 48]}</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>{KnotDet[Knot[11, Alternating, 48]], KnotSignature[Knot[11, Alternating, 48]]}</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>{113, -4}</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>J=Jones[Knot[11, Alternating, 48]][q]</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>     -11    3    7    12   16   18   18   16   11   7    3
1 - q    + --- - -- + -- - -- + -- - -- + -- - -- + -- - -
            10    9    8    7    6    5    4    3    2   q
           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[11]:=</nowiki></pre></td><td><pre style="color: red; border: 0px; padding: 0em"><nowiki>Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]</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>{Knot[11, Alternating, 48]}</nowiki></pre></td></tr>
<math>\textrm{Include}(\textrm{ColouredJonesM.mhtml})</math>
<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>A2Invariant[Knot[11, Alternating, 48]][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>     -34    -30    3     2     -24    2     3     3     2     -12
1 - q    + q    - --- + --- + q    - --- + --- - --- + --- - q    + 
                   28    26           22    20    18    16
                  q     q            q     q     q     q
 
   4    3    2     -4    -2
  --- - -- + -- + q   - q
   10    8    6
  q     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>Kauffman[Knot[11, Alternating, 48]][a, z]</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>  2    8    10    3        5        7        9      11      13
-a  + a  + a   - a  z - 3 a  z - 2 a  z - 2 a  z - a   z + a   z + 
 
     2  2      4  2      6  2       8  2      10  2      12  2
  3 a  z  + 2 a  z  - 4 a  z  - 10 a  z  - 5 a   z  + 2 a   z  + 
 
     3  3       5  3      7  3      9  3      11  3      13  3
  6 a  z  + 11 a  z  + 7 a  z  + 7 a  z  + 3 a   z  - 2 a   z  - 
 
     2  4       6  4       8  4      10  4      12  4      3  5
  3 a  z  + 16 a  z  + 26 a  z  + 8 a   z  - 5 a   z  - 8 a  z  - 
 
      5  5      7  5      9  5      11  5    13  5    2  6      4  6
  14 a  z  - 5 a  z  - 7 a  z  - 7 a   z  + a   z  + a  z  - 7 a  z  - 
 
      6  6       8  6      10  6      12  6      3  7    5  7
  23 a  z  - 27 a  z  - 9 a   z  + 3 a   z  + 3 a  z  + a  z  - 
 
     7  7      11  7      4  8      6  8       8  8      10  8
  7 a  z  + 5 a   z  + 4 a  z  + 9 a  z  + 11 a  z  + 6 a   z  + 
 
     5  9      7  9      9  9    6  10    8  10
  3 a  z  + 7 a  z  + 4 a  z  + a  z   + a  z</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>{Vassiliev[2][Knot[11, Alternating, 48]], Vassiliev[3][Knot[11, Alternating, 48]]}</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>{0, -10}</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>Kh[Knot[11, Alternating, 48]][q, t]</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>3    5      1        2        1        5        2        7
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + 
 5    3    23  9    21  8    19  8    19  7    17  7    17  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t
 
    5        9        7        9        9        9        9       7
  ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
   15  6    15  5    13  5    13  4    11  4    11  3    9  3    9  2
  q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
    9      4      7     t    2 t      2
  ----- + ---- + ---- + -- + --- + q t
   7  2    7      5      3    q
  q  t    q  t   q  t   q</nowiki></pre></td></tr>
</table>

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