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Acknowledgement

L7a6 as Morse Link
DrawMorseLink

PD Presentation: X8192 X10,4,11,3 X14,10,7,9 X12,6,13,5 X2738 X4,12,5,11 X6,14,1,13

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

Jones Polynomial: - q-1/2 + q1/2 - 2q3/2 + 2q5/2 - 3q7/2 + 2q9/2 - 2q11/2 + q13/2

A2 (sl(3)) Invariant: q-2 + 1 + q2 + q4 + 2q8 + q10 + 2q12 + q14 - q20

HOMFLY-PT Polynomial: 2a-5z + a-5z3 - a-3z-1 - 4a-3z - 4a-3z3 - a-3z5 + a-1z-1 + 3a-1z + a-1z3

Kauffman Polynomial: - a-8z2 + a-7z - 2a-7z3 + a-6z2 - 2a-6z4 - a-5z + 3a-5z3 - 2a-5z5 + a-4z2 + a-4z4 - a-4z6 + a-3z-1 - 6a-3z + 9a-3z3 - 3a-3z5 - a-2 - a-2z2 + 3a-2z4 - a-2z6 + a-1z-1 - 4a-1z + 4a-1z3 - a-1z5

Khovanov Homology:
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 14       1
j = 12      1 
j = 10     11 
j = 8    21  
j = 6   12   
j = 4  11    
j = 2 12     
j = 0        
j = -21       


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


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L7a6
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