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L10n76

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Acknowledgement

L10n76 as Morse Link
DrawMorseLink

PD Presentation: X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,12,20,5 X11,20,12,13 X13,10,14,11

Gauss Code: {{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 10, -9, 8}, {-10, 2, 4, -5, 7, 6, -8, 9}}

Jones Polynomial: q-9 - 2q-8 + 4q-7 - 4q-6 + 6q-5 - 5q-4 + 5q-3 - 3q-2 + 2q-1

A2 (sl(3)) Invariant: q-28 + q-24 + 4q-22 + 3q-20 + 5q-18 + 4q-16 + 2q-14 + 2q-12 + 2q-8 + q-6 + 2q-2

HOMFLY-PT Polynomial: 2a2 + 2a2z2 + a4z-2 - a4z2 - a4z4 - 2a6z-2 - 3a6 - 2a6z2 - a6z4 + a8z-2 + a8 + a8z2

Kauffman Polynomial: - 2a2 + 3a2z2 + a3z + a3z3 + a3z5 - a4z-2 + a4 + 2a4z2 - 2a4z4 + 2a4z6 + 2a5z-1 - 6a5z + 4a5z3 - 3a5z5 + 2a5z7 - 2a6z-2 + 5a6 - 6a6z4 + a6z6 + a6z8 + 2a7z-1 - 6a7z + 8a7z3 - 11a7z5 + 4a7z7 - a8z-2 + a8 + 6a8z2 - 8a8z4 + a8z8 + a9z + 5a9z3 - 7a9z5 + 2a9z7 - 2a10 + 5a10z2 - 4a10z4 + a10z6

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -1        2
j = -3       32
j = -5      2  
j = -7     33  
j = -9    32   
j = -11   13    
j = -13  33     
j = -15 13      
j = -17 1       
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[10, NonAlternating, 76]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 76]]
Out[4]=   
PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[8, 4, 9, 3], X[2, 16, 3, 15], 
 
>   X[16, 7, 17, 8], X[9, 18, 10, 19], X[4, 17, 1, 18], X[19, 12, 20, 5], 
 
>   X[11, 20, 12, 13], X[13, 10, 14, 11]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, 3, -7}, {-2, -1, 5, -3, -6, 10, -9, 8}, 
 
>   {-10, 2, 4, -5, 7, 6, -8, 9}]
In[6]:=
Jones[L][q]
Out[6]=   
 -9   2    4    4    6    5    5    3    2
q   - -- + -- - -- + -- - -- + -- - -- + -
       8    7    6    5    4    3    2   q
      q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -28    -24    4     3     5     4     2     2    2     -6   2
q    + q    + --- + --- + --- + --- + --- + --- + -- + q   + --
               22    20    18    16    14    12    8          2
              q     q     q     q     q     q     q          q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 76]][a, z]
Out[8]=   
                    4      6    8
   2      6    8   a    2 a    a       2  2    4  2      6  2    8  2    4  4
2 a  - 3 a  + a  + -- - ---- + -- + 2 a  z  - a  z  - 2 a  z  + a  z  - a  z  - 
                    2     2     2
                   z     z     z
 
     6  4
>   a  z
In[9]:=
Kauffman[Link[10, NonAlternating, 76]][a, z]
Out[9]=   
                                  4      6    8      5      7
    2    4      6    8      10   a    2 a    a    2 a    2 a     3        5
-2 a  + a  + 5 a  + a  - 2 a   - -- - ---- - -- + ---- + ---- + a  z - 6 a  z - 
                                  2     2     2    z      z
                                 z     z     z
 
       7      9        2  2      4  2      8  2      10  2    3  3      5  3
>   6 a  z + a  z + 3 a  z  + 2 a  z  + 6 a  z  + 5 a   z  + a  z  + 4 a  z  + 
 
       7  3      9  3      4  4      6  4      8  4      10  4    3  5
>   8 a  z  + 5 a  z  - 2 a  z  - 6 a  z  - 8 a  z  - 4 a   z  + a  z  - 
 
       5  5       7  5      9  5      4  6    6  6    10  6      5  7
>   3 a  z  - 11 a  z  - 7 a  z  + 2 a  z  + a  z  + a   z  + 2 a  z  + 
 
       7  7      9  7    6  8    8  8
>   4 a  z  + 2 a  z  + a  z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    2     1        1        1        3        3        3        1
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 3   q    19  8    17  7    15  7    15  6    13  6    13  5    11  5
q        q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        3       2       3       3       2      3
>   ------ + ----- + ----- + ----- + ----- + ----- + ----
     11  4    9  4    9  3    7  3    7  2    5  2    3
    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 L10n76
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