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L9a32

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Acknowledgement

L9a32 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-21/2 - 3q-19/2 + 5q-17/2 - 7q-15/2 + 8q-13/2 - 9q-11/2 + 7q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant: - q-34 - 2q-32 + q-30 + 4q-24 + q-22 + 2q-20 + q-18 + 2q-14 - q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial: - a3z3 - 3a5z - 3a5z3 - 2a7z-1 - 4a7z - 3a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial: - a3z3 - 3a4z4 - 3a5z + 6a5z3 - 6a5z5 - 3a6z2 + 9a6z4 - 7a6z6 - 2a7z-1 + 6a7z - 6a7z3 + 10a7z5 - 6a7z7 + 3a8 - 14a8z2 + 21a8z4 - 5a8z6 - 2a8z8 - 3a9z-1 + 11a9z - 22a9z3 + 26a9z5 - 9a9z7 + 3a10 - 14a10z2 + 12a10z4 + a10z6 - 2a10z8 - a11z-1 + 2a11z - 9a11z3 + 10a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2         1
j = -4        31
j = -6       3  
j = -8      43  
j = -10     53   
j = -12    34    
j = -14   45     
j = -16  24      
j = -18 13       
j = -20 2        
j = -221         


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[9, Alternating, 32]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[9, Alternating, 32]]
Out[4]=   
PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 13, 7, 14], X[14, 9, 15, 10], 
 
>   X[10, 17, 11, 18], X[16, 5, 17, 6], X[2, 7, 3, 8], X[4, 11, 5, 12], 
 
>   X[6, 15, 1, 16]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -7, 2, -8, 6, -9}, {7, -1, 4, -5, 8, -2, 3, -4, 9, -6, 5, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(21/2)     3       5       7       8       9      7      6      3      -(3/2)
q        - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - q
            19/2    17/2    15/2    13/2    11/2    9/2    7/2    5/2
           q       q       q       q       q       q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -34    2     -30    4     -22    2     -18    2     -12    2     -8   2     -4
-q    - --- + q    + --- + q    + --- + q    + --- - q    + --- + q   - -- + q
         32           24           20           14           10          6
        q            q            q            q            q           q
In[8]:=
HOMFLYPT[Link[9, Alternating, 32]][a, z]
Out[8]=   
    7      9    11
-2 a    3 a    a        5        7        9      3  3      5  3      7  3
----- + ---- - --- - 3 a  z - 4 a  z + 4 a  z - a  z  - 3 a  z  - 3 a  z
  z      z      z
In[9]:=
Kauffman[Link[9, Alternating, 32]][a, z]
Out[9]=   
                        7      9    11
   8      10    12   2 a    3 a    a        5        7         9        11
3 a  + 3 a   + a   - ---- - ---- - --- - 3 a  z + 6 a  z + 11 a  z + 2 a   z - 
                      z      z      z
 
       6  2       8  2       10  2      12  2    3  3      5  3      7  3
>   3 a  z  - 14 a  z  - 14 a   z  - 3 a   z  - a  z  + 6 a  z  - 6 a  z  - 
 
        9  3      11  3      4  4      6  4       8  4       10  4      12  4
>   22 a  z  - 9 a   z  - 3 a  z  + 9 a  z  + 21 a  z  + 12 a   z  + 3 a   z  - 
 
       5  5       7  5       9  5       11  5      6  6      8  6    10  6
>   6 a  z  + 10 a  z  + 26 a  z  + 10 a   z  - 7 a  z  - 5 a  z  + a   z  - 
 
     12  6      7  7      9  7      11  7      8  8      10  8
>   a   z  - 6 a  z  - 9 a  z  - 3 a   z  - 2 a  z  - 2 a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -4    -2     1        2        1        3        2        4        4
q   + q   + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
             22  9    20  8    18  8    18  7    16  7    16  6    14  6
            q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      5        3        4        5        3        4       3       3      3
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----
     14  5    12  5    12  4    10  4    10  3    8  3    8  2    6  2    4
    q   t    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t


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