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Acknowledgement

L8n6 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X11,16,12,13 X7,14,8,15 X13,8,14,9 X15,12,16,5 X2536 X4,9,1,10

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

Jones Polynomial: q-9 + q-7 + q-6 + q-2

A2 (sl(3)) Invariant: q-30 + 3q-28 + 4q-26 + 5q-24 + 4q-22 + 3q-20 + 2q-18 + q-16 + q-14 + q-10 + q-8 + q-6

HOMFLY-PT Polynomial: 2a4 + 4a4z2 + a4z4 + a6z-2 - 2a8z-2 - 2a8 + a10z-2

Kauffman Polynomial: 2a4 - 4a4z2 + a4z4 + a6z-2 - 2a6 - 2a7z-1 + 8a7z - 6a7z3 + a7z5 + 2a8z-2 - 9a8 + 14a8z2 - 7a8z4 + a8z6 - 2a9z-1 + 8a9z - 6a9z3 + a9z5 + a10z-2 - 6a10 + 10a10z2 - 6a10z4 + a10z6

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -3        1
j = -5        1
j = -7     11  
j = -9         
j = -11   131   
j = -13    2    
j = -15  1      
j = -171        
j = -191        

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]=   
3
In[3]:=
Show[DrawMorseLink[Link[8, NonAlternating, 6]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[8, NonAlternating, 6]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 16, 12, 13], X[7, 14, 8, 15], 
 
>   X[13, 8, 14, 9], X[15, 12, 16, 5], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -7, 2, -8}, {-5, 4, -6, 3}, {7, -1, -4, 5, 8, -2, -3, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -9    -7    -6    -2
q   + q   + q   + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -30    3     4     5     4     3     2     -16    -14    -10    -8    -6
q    + --- + --- + --- + --- + --- + --- + q    + q    + q    + q   + q
        28    26    24    22    20    18
       q     q     q     q     q     q
In[8]:=
HOMFLYPT[Link[8, NonAlternating, 6]][a, z]
Out[8]=   
               6      8    10
   4      8   a    2 a    a        4  2    4  4
2 a  - 2 a  + -- - ---- + --- + 4 a  z  + a  z
               2     2     2
              z     z     z
In[9]:=
Kauffman[Link[8, NonAlternating, 6]][a, z]
Out[9]=   
                              6      8    10      7      9
   4      6      8      10   a    2 a    a     2 a    2 a       7        9
2 a  - 2 a  - 9 a  - 6 a   + -- + ---- + --- - ---- - ---- + 8 a  z + 8 a  z - 
                              2     2     2     z      z
                             z     z     z
 
       4  2       8  2       10  2      7  3      9  3    4  4      8  4
>   4 a  z  + 14 a  z  + 10 a   z  - 6 a  z  - 6 a  z  + a  z  - 7 a  z  - 
 
       10  4    7  5    9  5    8  6    10  6
>   6 a   z  + a  z  + a  z  + a  z  + a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -5    -3     1        1        1        1        2        3        1
q   + q   + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
             19  8    17  8    15  6    11  5    13  4    11  4    11  3
            q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      1       1
>   ----- + -----
     7  3    7  2
    q  t    q  t

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