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L10n36

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Acknowledgement

L10n36 as Morse Link
DrawMorseLink

PD Presentation: X6172 X3,13,4,12 X16,13,17,14 X14,9,15,10 X10,15,11,16 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X11,1,12,4

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

Jones Polynomial: q-9/2 - q-7/2 + q-5/2 - q-3/2 - q-1/2 - q1/2 - q3/2 + q5/2 - q7/2 + q9/2

A2 (sl(3)) Invariant: - q-14 - q-12 - 2q-10 - q-8 + 2q-4 + 5q-2 + 5 + 5q2 + 2q4 - q8 - 2q10 - q12 - q14

HOMFLY-PT Polynomial: 2a-3z-1 + 3a-3z + a-3z3 - 7a-1z-1 - 11a-1z - 6a-1z3 - a-1z5 + 7az-1 + 11az + 6az3 + az5 - 2a3z-1 - 3a3z - a3z3

Kauffman Polynomial: 2a-4 - 6a-4z2 + 5a-4z4 - a-4z6 - 2a-3z-1 + 4a-3z - 6a-3z3 + 5a-3z5 - a-3z7 + 8a-2 - 18a-2z2 + 12a-2z4 - 2a-2z6 - 7a-1z-1 + 14a-1z - 14a-1z3 + 7a-1z5 - a-1z7 + 13 - 24z2 + 14z4 - 2z6 - 7az-1 + 14az - 14az3 + 7az5 - az7 + 8a2 - 18a2z2 + 12a2z4 - 2a2z6 - 2a3z-1 + 4a3z - 6a3z3 + 5a3z5 - a3z7 + 2a4 - 6a4z2 + 5a4z4 - a4z6

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 10          1
j = 8           
j = 6        11 
j = 4      11   
j = 2     1 1   
j = 0    141    
j = -2   1 1     
j = -4   11      
j = -6 11        
j = -8           
j = -101          


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[10, NonAlternating, 36]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 36]]
Out[4]=   
PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[16, 13, 17, 14], X[14, 9, 15, 10], 
 
>   X[10, 15, 11, 16], X[17, 5, 18, 20], X[7, 19, 8, 18], X[19, 9, 20, 8], 
 
>   X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, -2, 10}, {9, -1, -7, 8, 4, -5, -10, 2, 3, -4, 5, -3, -6, 7, 
 
>    -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(9/2)    -(7/2)    -(5/2)    -(3/2)      1                 3/2    5/2
q       - q       + q       - q       - ------- - Sqrt[q] - q    + q    - 
                                        Sqrt[q]
 
     7/2    9/2
>   q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -14    -12    2     -8   2    5       2      4    8      10    12    14
5 - q    - q    - --- - q   + -- + -- + 5 q  + 2 q  - q  - 2 q   - q   - q
                   10          4    2
                  q           q    q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 36]][a, z]
Out[8]=   
                      3                                   3      3
 2      7    7 a   2 a    3 z   11 z               3     z    6 z         3
---- - --- + --- - ---- + --- - ---- + 11 a z - 3 a  z + -- - ---- + 6 a z  - 
 3     a z    z     z      3     a                        3    a
a  z                      a                              a
 
             5
     3  3   z       5
>   a  z  - -- + a z
            a
In[9]:=
Kauffman[Link[10, NonAlternating, 36]][a, z]
Out[9]=   
                                                   3
     2    8       2      4    2      7    7 a   2 a    4 z   14 z
13 + -- + -- + 8 a  + 2 a  - ---- - --- - --- - ---- + --- + ---- + 14 a z + 
      4    2                  3     a z    z     z      3     a
     a    a                  a  z                      a
 
                        2       2                           3       3
       3         2   6 z    18 z        2  2      4  2   6 z    14 z
>   4 a  z - 24 z  - ---- - ----- - 18 a  z  - 6 a  z  - ---- - ----- - 
                       4      2                            3      a
                      a      a                            a
 
                                   4       4                           5
          3      3  3       4   5 z    12 z        2  4      4  4   5 z
>   14 a z  - 6 a  z  + 14 z  + ---- + ----- + 12 a  z  + 5 a  z  + ---- + 
                                  4      2                            3
                                 a      a                            a
 
       5                              6      6                      7    7
    7 z         5      3  5      6   z    2 z       2  6    4  6   z    z
>   ---- + 7 a z  + 5 a  z  - 2 z  - -- - ---- - 2 a  z  - a  z  - -- - -- - 
     a                                4     2                       3   a
                                     a     a                       a
 
       7    3  7
>   a z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     -2    2     1        1       1       1       1     1    1          4
4 + q   + q  + ------ + ----- + ----- + ----- + ----- + - + ---- + t + q  t + 
                10  5    6  4    6  3    4  2    2  2   t    4
               q   t    q  t    q  t    q  t    q  t        q  t
 
     2  2    4  2    6  3    6  4    10  5
>   q  t  + q  t  + q  t  + q  t  + q   t


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