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L11n11

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Acknowledgement

L11n11 as Morse Link
DrawMorseLink

PD Presentation: X6172 X16,7,17,8 X17,1,18,4 X5,10,6,11 X3849 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X2,16,3,15

Gauss Code: {{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, 11, -2, -3, 10, -9, 7, -8, 6}}

Jones Polynomial: q-21/2 - 2q-19/2 + 4q-17/2 - 5q-15/2 + 6q-13/2 - 7q-11/2 + 6q-9/2 - 5q-7/2 + 2q-5/2 - 2q-3/2

A2 (sl(3)) Invariant: - q-32 - q-28 - 2q-26 - 2q-22 + q-18 + q-16 + 3q-14 + q-12 + 3q-10 + 3q-8 + q-6 + 2q-4

HOMFLY-PT Polynomial: - 2a3z-1 - 5a3z - 2a3z3 + 2a5z-1 + 4a5z + 3a5z3 + a5z5 + a7z-1 + 3a7z + 3a7z3 + a7z5 - a9z-1 - 2a9z - a9z3

Kauffman Polynomial: - 2a3z-1 + 7a3z - 3a3z3 + 3a4 - 4a4z2 + 2a4z4 - a4z6 - 2a5z-1 + 8a5z - 13a5z3 + 7a5z5 - 2a5z7 + 3a6z2 - 11a6z4 + 7a6z6 - 2a6z8 + a7z-1 - 8a7z3 + 4a7z5 + a7z7 - a7z9 - 3a8 + 12a8z2 - 20a8z4 + 15a8z6 - 4a8z8 + a9z-1 - 3a9z3 + 4a9z5 + a9z7 - a9z9 + a10z2 - 3a10z4 + 6a10z6 - 2a10z8 + a11z - 5a11z3 + 7a11z5 - 2a11z7 + a12 - 4a12z2 + 4a12z4 - a12z6

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2         2
j = -4        22
j = -6       3  
j = -8      32  
j = -10     43   
j = -12    23    
j = -14   34     
j = -16  12      
j = -18 13       
j = -20 1        
j = -221         


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[11, NonAlternating, 11]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 11]]
Out[4]=   
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[5, 10, 6, 11], 
 
>   X[3, 8, 4, 9], X[13, 22, 14, 5], X[11, 20, 12, 21], X[21, 12, 22, 13], 
 
>   X[19, 14, 20, 15], X[9, 18, 10, 19], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, 11, -2, -3, 10, 
 
>    -9, 7, -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(21/2)     2       4       5       6       7      6      5      2      2
q        - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ----
            19/2    17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
           q       q       q       q       q       q      q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -32    -28    2     2     -18    -16    3     -12    3    3     -6   2
-q    - q    - --- - --- + q    + q    + --- + q    + --- + -- + q   + --
                26    22                  14           10    8          4
               q     q                   q            q     q          q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 11]][a, z]
Out[8]=   
    3      5    7    9
-2 a    2 a    a    a       3        5        7        9        3  3
----- + ---- + -- - -- - 5 a  z + 4 a  z + 3 a  z - 2 a  z - 2 a  z  + 
  z      z     z    z
 
       5  3      7  3    9  3    5  5    7  5
>   3 a  z  + 3 a  z  - a  z  + a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 11]][a, z]
Out[9]=   
                       3      5    7    9
   4      8    12   2 a    2 a    a    a       3        5      11        4  2
3 a  - 3 a  + a   - ---- - ---- + -- + -- + 7 a  z + 8 a  z + a   z - 4 a  z  + 
                     z      z     z    z
 
       6  2       8  2    10  2      12  2      3  3       5  3      7  3
>   3 a  z  + 12 a  z  + a   z  - 4 a   z  - 3 a  z  - 13 a  z  - 8 a  z  - 
 
       9  3      11  3      4  4       6  4       8  4      10  4      12  4
>   3 a  z  - 5 a   z  + 2 a  z  - 11 a  z  - 20 a  z  - 3 a   z  + 4 a   z  + 
 
       5  5      7  5      9  5      11  5    4  6      6  6       8  6
>   7 a  z  + 4 a  z  + 4 a  z  + 7 a   z  - a  z  + 7 a  z  + 15 a  z  + 
 
       10  6    12  6      5  7    7  7    9  7      11  7      6  8
>   6 a   z  - a   z  - 2 a  z  + a  z  + a  z  - 2 a   z  - 2 a  z  - 
 
       8  8      10  8    7  9    9  9
>   4 a  z  - 2 a   z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    2      1        1        1        3        1        2        3
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    22  9    20  8    18  8    18  7    16  7    16  6    14  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      4        2        3        4        3        3       2       3      2
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----
     14  5    12  5    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    q  t    q  t


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