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L11a243

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Acknowledgement

L11a243 as Morse Link
DrawMorseLink

PD Presentation: X8192 X12,4,13,3 X22,10,7,9 X16,11,17,12 X10,15,11,16 X2738 X20,17,21,18 X6,14,1,13 X4,20,5,19 X18,6,19,5 X14,21,15,22

Gauss Code: {{1, -6, 2, -9, 10, -8}, {6, -1, 3, -5, 4, -2, 8, -11, 5, -4, 7, -10, 9, -7, 11, -3}}

Jones Polynomial: - q-13/2 + 4q-11/2 - 9q-9/2 + 16q-7/2 - 21q-5/2 + 24q-3/2 - 25q-1/2 + 20q1/2 - 16q3/2 + 9q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant: q-20 - q-18 - q-16 + 3q-14 - 5q-12 + q-10 - 4q-6 + 5q-4 - 2q-2 + 7 + 2q2 + 5q6 - 3q8 + q10 + q12 - q14

HOMFLY-PT Polynomial: a-3z + a-3z3 - 2a-1z-1 - 3a-1z - 4a-1z3 - 2a-1z5 + 3az-1 + 5az + 5az3 + 3az5 + az7 - a3z-1 - 3a3z - 4a3z3 - 2a3z5 + a5z + a5z3

Kauffman Polynomial: - a-4z2 + 2a-4z4 - a-4z6 + 2a-3z - 7a-3z3 + 9a-3z5 - 4a-3z7 - 7a-2z4 + 14a-2z6 - 7a-2z8 - 2a-1z-1 + 6a-1z - 15a-1z3 + 15a-1z5 + 3a-1z7 - 6a-1z9 + 3 + 2z2 - 24z4 + 39z6 - 15z8 - 2z10 - 3az-1 + 5az - 12az3 + 10az5 + 14az7 - 13az9 + 3a2 + 3a2z2 - 26a2z4 + 41a2z6 - 18a2z8 - 2a2z10 - a3z-1 + 2a3z - 11a3z3 + 16a3z5 - a3z7 - 7a3z9 + a4 - 6a4z4 + 13a4z6 - 10a4z8 + a5z - 6a5z3 + 11a5z5 - 8a5z7 - 2a6z2 + 5a6z4 - 4a6z6 + a7z3 - a7z5

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 10           1
j = 8          3 
j = 6         61 
j = 4        103  
j = 2       117   
j = 0      149    
j = -2     1112     
j = -4    1013      
j = -6   611       
j = -8  310        
j = -10 16         
j = -12 3          
j = -141           


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
Length[Skeleton[L]]
Out[2]=   
2
In[3]:=
Show[DrawMorseLink[Link[11, Alternating, 243]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 243]]
Out[4]=   
PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[22, 10, 7, 9], X[16, 11, 17, 12], 
 
>   X[10, 15, 11, 16], X[2, 7, 3, 8], X[20, 17, 21, 18], X[6, 14, 1, 13], 
 
>   X[4, 20, 5, 19], X[18, 6, 19, 5], X[14, 21, 15, 22]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -6, 2, -9, 10, -8}, 
 
>   {6, -1, 3, -5, 4, -2, 8, -11, 5, -4, 7, -10, 9, -7, 11, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(13/2)     4      9      16     21     24      25
-q        + ----- - ---- + ---- - ---- + ---- - ------- + 20 Sqrt[q] - 
             11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q      q      q      q
 
        3/2      5/2      7/2    9/2
>   16 q    + 9 q    - 4 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -20    -18    -16    3     5     -10   4    5    2       2      6      8
7 + q    - q    - q    + --- - --- + q    - -- + -- - -- + 2 q  + 5 q  - 3 q  + 
                          14    12           6    4    2
                         q     q            q    q    q
 
     10    12    14
>   q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 243]][a, z]
Out[8]=   
             3                                       3      3
-2    3 a   a    z    3 z              3      5     z    4 z         3
--- + --- - -- + -- - --- + 5 a z - 3 a  z + a  z + -- - ---- + 5 a z  - 
a z    z    z     3    a                             3    a
                 a                                  a
 
                         5
       3  3    5  3   2 z         5      3  5      7
>   4 a  z  + a  z  - ---- + 3 a z  - 2 a  z  + a z
                       a
In[9]:=
Kauffman[Link[11, Alternating, 243]][a, z]
Out[9]=   
                             3
       2    4    2    3 a   a    2 z   6 z              3      5        2
3 + 3 a  + a  - --- - --- - -- + --- + --- + 5 a z + 2 a  z + a  z + 2 z  - 
                a z    z    z     3     a
                                 a
 
     2                          3       3
    z       2  2      6  2   7 z    15 z          3       3  3      5  3
>   -- + 3 a  z  - 2 a  z  - ---- - ----- - 12 a z  - 11 a  z  - 6 a  z  + 
     4                         3      a
    a                         a
 
                       4      4                                     5       5
     7  3       4   2 z    7 z        2  4      4  4      6  4   9 z    15 z
>   a  z  - 24 z  + ---- - ---- - 26 a  z  - 6 a  z  + 5 a  z  + ---- + ----- + 
                      4      2                                     3      a
                     a      a                                     a
 
                                                     6       6
          5       3  5       5  5    7  5       6   z    14 z        2  6
>   10 a z  + 16 a  z  + 11 a  z  - a  z  + 39 z  - -- + ----- + 41 a  z  + 
                                                     4     2
                                                    a     a
 
                            7      7
        4  6      6  6   4 z    3 z          7    3  7      5  7       8
>   13 a  z  - 4 a  z  - ---- + ---- + 14 a z  - a  z  - 8 a  z  - 15 z  - 
                           3     a
                          a
 
       8                            9
    7 z        2  8       4  8   6 z          9      3  9      10      2  10
>   ---- - 18 a  z  - 10 a  z  - ---- - 13 a z  - 7 a  z  - 2 z   - 2 a  z
      2                           a
     a
In[10]:=
Kh[L][q, t]
Out[10]=   
     12     1        3        1        6        3      10       6      11
14 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
      2    14  6    12  5    10  5    10  4    8  4    8  3    6  3    6  2
     q    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
     10      13     11              2        2  2       4  2      4  3
>   ----- + ---- + ---- + 9 t + 11 q  t + 7 q  t  + 10 q  t  + 3 q  t  + 
     4  2    4      2
    q  t    q  t   q  t
 
       6  3    6  4      8  4    10  5
>   6 q  t  + q  t  + 3 q  t  + q   t


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