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Acknowledgement

L8a13 as Morse Link
DrawMorseLink

PD Presentation: X10,1,11,2 X16,7,9,8 X12,3,13,4 X6,13,7,14 X14,5,15,6 X4,15,5,16 X2,9,3,10 X8,11,1,12

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

Jones Polynomial: - q-19/2 + q-17/2 - 3q-15/2 + 4q-13/2 - 4q-11/2 + 4q-9/2 - 4q-7/2 + 2q-5/2 - q-3/2

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

HOMFLY-PT Polynomial: - a3z - a3z3 - 3a5z - 2a5z3 - a7z-1 - a7z - a7z3 + a9z-1 + a9z

Kauffman Polynomial: a3z - a3z3 + a4z2 - 2a4z4 - 3a5z + 4a5z3 - 3a5z5 + a6z4 - 2a6z6 - a7z-1 + 3a7z - a7z3 - a7z7 + a8 - a8z2 + 5a8z4 - 3a8z6 - a9z-1 + 3a9z - 2a9z3 + 2a9z5 - a9z7 + 2a10z4 - a10z6 - 4a11z + 4a11z3 - a11z5

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2        1
j = -4       21
j = -6      2  
j = -8     22  
j = -10    22   
j = -12   22    
j = -14  12     
j = -16  2      
j = -1811       
j = -201        


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


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L8a13
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