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L10n2

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Acknowledgement

L10n2 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,19,12,18 X17,5,18,20 X19,13,20,12 X9,16,10,17 X2,14,3,13

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

Jones Polynomial: q-9/2 - 2q-7/2 + 3q-5/2 - 5q-3/2 + 5q-1/2 - 6q1/2 + 4q3/2 - 4q5/2 + 2q7/2

A2 (sl(3)) Invariant: - q-14 - q-10 - q-8 + 2q-6 + q-4 + 3q-2 + 3 + 2q2 + 3q4 + q8 - q10 - 2q12

HOMFLY-PT Polynomial: a-3z-1 + 2a-3z - 4a-1z-1 - 7a-1z - 3a-1z3 + 4az-1 + 7az + 4az3 + az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial: a-4 - 3a-4z2 - a-3z-1 + 4a-3z - 3a-3z3 - a-3z5 + 4a-2 - 10a-2z2 + 7a-2z4 - 3a-2z6 - 4a-1z-1 + 13a-1z - 15a-1z3 + 9a-1z5 - 3a-1z7 + 7 - 15z2 + 13z4 - 2z6 - z8 - 4az-1 + 13az - 21az3 + 18az5 - 5az7 + 4a2 - 12a2z2 + 10a2z4 - a2z8 - a3z-1 + 4a3z - 9a3z3 + 8a3z5 - 2a3z7 + a4 - 4a4z2 + 4a4z4 - a4z6

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 8        2
j = 6       2 
j = 4      22 
j = 2     42  
j = 0    34   
j = -2   22    
j = -4  13     
j = -6 12      
j = -8 1       
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, 2]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 2]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[15, 1, 16, 4], X[5, 10, 6, 11], 
 
>   X[3, 8, 4, 9], X[11, 19, 12, 18], X[17, 5, 18, 20], X[19, 13, 20, 12], 
 
>   X[9, 16, 10, 17], X[2, 14, 3, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, -5, 3}, {-4, -1, 2, 5, -9, 4, -6, 8, 10, -2, -3, 9, -7, 6, 
 
>    -8, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(9/2)    2      3      5        5                     3/2      5/2      7/2
q       - ---- + ---- - ---- + ------- - 6 Sqrt[q] + 4 q    - 4 q    + 2 q
           7/2    5/2    3/2   Sqrt[q]
          q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -14    -10    -8   2     -4   3       2      4    8    10      12
3 - q    - q    - q   + -- + q   + -- + 2 q  + 3 q  + q  - q   - 2 q
                         6          2
                        q          q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 2]][a, z]
Out[8]=   
                    3                                   3
 1      4    4 a   a    2 z   7 z              3     3 z         3    3  3
---- - --- + --- - -- + --- - --- + 7 a z - 2 a  z - ---- + 4 a z  - a  z  + 
 3     a z    z    z     3     a                      a
a  z                    a
 
       5
>   a z
In[9]:=
Kauffman[Link[10, NonAlternating, 2]][a, z]
Out[9]=   
                                               3
     -4   4       2    4    1      4    4 a   a    4 z   13 z
7 + a   + -- + 4 a  + a  - ---- - --- - --- - -- + --- + ---- + 13 a z + 
           2                3     a z    z    z     3     a
          a                a  z                    a
 
                        2       2                           3       3
       3         2   3 z    10 z        2  2      4  2   3 z    15 z
>   4 a  z - 15 z  - ---- - ----- - 12 a  z  - 4 a  z  - ---- - ----- - 
                       4      2                            3      a
                      a      a                            a
 
                                   4                         5      5
          3      3  3       4   7 z        2  4      4  4   z    9 z
>   21 a z  - 9 a  z  + 13 z  + ---- + 10 a  z  + 4 a  z  - -- + ---- + 
                                  2                          3    a
                                 a                          a
 
                                  6              7
          5      3  5      6   3 z     4  6   3 z         7      3  7    8
>   18 a z  + 8 a  z  - 2 z  - ---- - a  z  - ---- - 5 a z  - 2 a  z  - z  - 
                                 2             a
                                a
 
     2  8
>   a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
       2     1        1       1       2       1       3       2     3    2
4 + 4 q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 
            10  5    8  4    6  4    6  3    4  3    4  2    2  2   t    2
           q   t    q  t    q  t    q  t    q  t    q  t    q  t        q  t
 
       2        4        4  2      6  2      8  3
>   2 q  t + 2 q  t + 2 q  t  + 2 q  t  + 2 q  t


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