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Acknowledgement

L9a9 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X2,14,3,13

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

Jones Polynomial: q-11/2 - 3q-9/2 + 5q-7/2 - 7q-5/2 + 8q-3/2 - 9q-1/2 + 6q1/2 - 5q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant: - q-16 + q-14 - q-12 + q-10 + q-8 - q-6 + 3q-4 + 4 + q2 + q6 - q8 + q10

HOMFLY-PT Polynomial: - a-1z-1 - 2a-1z - 3a-1z3 - a-1z5 + az-1 + 4az + 8az3 + 5az5 + az7 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial: 2a-3z3 - a-3z5 - 2a-2z2 + 7a-2z4 - 3a-2z6 - a-1z-1 + 4a-1z - 10a-1z3 + 11a-1z5 - 4a-1z7 + 1 - 3z2 + 4z4 + z6 - 2z8 - az-1 + 8az - 24az3 + 22az5 - 8az7 - 3a2z2 + 3a2z4 - 2a2z8 + 4a3z - 8a3z3 + 7a3z5 - 4a3z7 - a4z2 + 5a4z4 - 4a4z6 + 4a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8         1
j = 6        2 
j = 4       31 
j = 2      32  
j = 0     63   
j = -2    45    
j = -4   34     
j = -6  24      
j = -8 13       
j = -10 2        
j = -121         


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


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