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Acknowledgement

L9n6 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X9,16,10,17 X11,18,12,5 X17,10,18,11 X2,14,3,13

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

Jones Polynomial: q-17/2 - 2q-15/2 + 4q-13/2 - 4q-11/2 + 4q-9/2 - 5q-7/2 + 2q-5/2 - 2q-3/2

A2 (sl(3)) Invariant: - q-26 - 2q-22 - 3q-20 - q-18 - q-16 + 3q-14 + 3q-12 + 4q-10 + 4q-8 + q-6 + 2q-4

HOMFLY-PT Polynomial: - 3a3z-1 - 5a3z - 2a3z3 + 5a5z-1 + 7a5z + 4a5z3 + a5z5 - 2a7z-1 - 2a7z - a7z3

Kauffman Polynomial: - 3a3z-1 + 6a3z - 3a3z3 + 5a4 - 6a4z2 + 2a4z4 - a4z6 - 5a5z-1 + 9a5z - 7a5z3 + 2a5z5 - a5z7 + 5a6 - 7a6z2 + 5a6z4 - 3a6z6 - 2a7z-1 + 3a7z - a7z3 - a7z7 + a8z2 + 2a8z4 - 2a8z6 + 3a9z3 - 2a9z5 - a10 + 2a10z2 - a10z4

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2       2
j = -4      22
j = -6     3  
j = -8    12  
j = -10   33   
j = -12  11    
j = -14 13     
j = -16 1      
j = -181       


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


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