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Acknowledgement

L11a33 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X18,11,19,12 X16,7,17,8 X8,17,9,18 X20,13,21,14 X22,15,5,16 X12,19,13,20 X14,21,15,22 X2536 X4,9,1,10

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

Jones Polynomial: q-29/2 - 2q-27/2 + 5q-25/2 - 7q-23/2 + 9q-21/2 - 11q-19/2 + 10q-17/2 - 9q-15/2 + 6q-13/2 - 5q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant: - q-44 - 2q-42 - q-40 - 3q-38 - q-36 - q-34 + q-32 + 4q-30 + q-28 + 5q-26 + q-24 + 2q-22 + 2q-20 + 2q-16 - q-14 + q-12

HOMFLY-PT Polynomial: - 3a7z - 7a7z3 - 5a7z5 - a7z7 - 4a9z-1 - 13a9z - 19a9z3 - 11a9z5 - 2a9z7 + 7a11z-1 + 18a11z + 14a11z3 + 3a11z5 - 3a13z-1 - 4a13z - a13z3

Kauffman Polynomial: 3a7z - 7a7z3 + 5a7z5 - a7z7 + a8z2 - 7a8z4 + 8a8z6 - 2a8z8 + 4a9z-1 - 16a9z + 24a9z3 - 23a9z5 + 14a9z7 - 3a9z9 - 7a10 + 26a10z2 - 39a10z4 + 23a10z6 - 2a10z8 - a10z10 + 7a11z-1 - 31a11z + 53a11z3 - 49a11z5 + 27a11z7 - 6a11z9 - 7a12 + 29a12z2 - 39a12z4 + 22a12z6 - 3a12z8 - a12z10 + 3a13z-1 - 12a13z + 20a13z3 - 17a13z5 + 9a13z7 - 3a13z9 + a14z2 - 3a14z4 + 4a14z6 - 3a14z8 + 2a15z5 - 3a15z7 - a16z2 + 3a16z4 - 3a16z6 + 2a17z3 - 2a17z5 - a18 + 2a18z2 - a18z4

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -6           1
j = -8          21
j = -10         3  
j = -12        32  
j = -14       63   
j = -16      54    
j = -18     65     
j = -20    35      
j = -22   46       
j = -24  13        
j = -26 14         
j = -28 1          
j = -301           


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, 33]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 33]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[16, 7, 17, 8], 
 
>   X[8, 17, 9, 18], X[20, 13, 21, 14], X[22, 15, 5, 16], X[12, 19, 13, 20], 
 
>   X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -8, 6, -9, 7, -4, 5, -3, 
 
>    8, -6, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(29/2)     2       5       7       9      11      10       9       6
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 
            27/2    25/2    23/2    21/2    19/2    17/2    15/2    13/2
           q       q       q       q       q       q       q       q
 
      5      2      -(7/2)
>   ----- + ---- - q
     11/2    9/2
    q       q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -44    2     -40    3     -36    -34    -32    4     -28    5     -24    2
-q    - --- - q    - --- - q    - q    + q    + --- + q    + --- + q    + --- + 
         42           38                         30           26           22
        q            q                          q            q            q
 
     2     2     -14    -12
>   --- + --- - q    + q
     20    16
    q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 33]][a, z]
Out[8]=   
    9      11      13
-4 a    7 a     3 a        7         9         11        13        7  3
----- + ----- - ----- - 3 a  z - 13 a  z + 18 a   z - 4 a   z - 7 a  z  - 
  z       z       z
 
        9  3       11  3    13  3      7  5       9  5      11  5    7  7
>   19 a  z  + 14 a   z  - a   z  - 5 a  z  - 11 a  z  + 3 a   z  - a  z  - 
 
       9  7
>   2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 33]][a, z]
Out[9]=   
                          9      11      13
    10      12    18   4 a    7 a     3 a        7         9         11
-7 a   - 7 a   - a   + ---- + ----- + ----- + 3 a  z - 16 a  z - 31 a   z - 
                        z       z       z
 
        13      8  2       10  2       12  2    14  2    16  2      18  2
>   12 a   z + a  z  + 26 a   z  + 29 a   z  + a   z  - a   z  + 2 a   z  - 
 
       7  3       9  3       11  3       13  3      17  3      8  4
>   7 a  z  + 24 a  z  + 53 a   z  + 20 a   z  + 2 a   z  - 7 a  z  - 
 
        10  4       12  4      14  4      16  4    18  4      7  5       9  5
>   39 a   z  - 39 a   z  - 3 a   z  + 3 a   z  - a   z  + 5 a  z  - 23 a  z  - 
 
        11  5       13  5      15  5      17  5      8  6       10  6
>   49 a   z  - 17 a   z  + 2 a   z  - 2 a   z  + 8 a  z  + 23 a   z  + 
 
        12  6      14  6      16  6    7  7       9  7       11  7      13  7
>   22 a   z  + 4 a   z  - 3 a   z  - a  z  + 14 a  z  + 27 a   z  + 9 a   z  - 
 
       15  7      8  8      10  8      12  8      14  8      9  9      11  9
>   3 a   z  - 2 a  z  - 2 a   z  - 3 a   z  - 3 a   z  - 3 a  z  - 6 a   z  - 
 
       13  9    10  10    12  10
>   3 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -8    -6      1         1         1        4        1        3        4
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             30  11    28  10    26  10    26  9    24  9    24  8    22  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      6        3        5        6        5        5        4        6
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     22  7    20  7    20  6    18  6    18  5    16  5    16  4    14  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        3        2        3       2
>   ------ + ------ + ------ + ------ + ----
     14  3    12  3    12  2    10  2    8
    q   t    q   t    q   t    q   t    q  t


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