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L11n20

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Acknowledgement

L11n20 as Morse Link
DrawMorseLink

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

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

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

A2 (sl(3)) Invariant: - q-32 - q-30 - 2q-28 - 3q-26 - q-22 + q-20 + 3q-18 + 2q-16 + 4q-14 + q-12 + 2q-10 + 2q-8 + 2q-4

HOMFLY-PT Polynomial: - a3z-1 - 4a3z - 2a3z3 - a5z-1 + 2a5z3 + a5z5 + 4a7z-1 + 7a7z + 4a7z3 + a7z5 - 2a9z-1 - 3a9z - a9z3

Kauffman Polynomial: - a3z-1 + 5a3z - 3a3z3 + a4 - a4z2 - a4z6 + a5z-1 - a5z - 5a5z3 + 4a5z5 - 2a5z7 - 5a6 + 15a6z2 - 17a6z4 + 7a6z6 - 2a6z8 + 4a7z-1 - 13a7z + 16a7z3 - 10a7z5 + 3a7z7 - a7z9 - 6a8 + 21a8z2 - 21a8z4 + 11a8z6 - 3a8z8 + 2a9z-1 - 8a9z + 17a9z3 - 11a9z5 + 4a9z7 - a9z9 + a10 - 2a10z2 + a10z4 + 2a10z6 - a10z8 - a11z - a11z3 + 3a11z5 - a11z7 + 2a12 - 7a12z2 + 5a12z4 - a12z6

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2         2
j = -4        32
j = -6       3  
j = -8      33  
j = -10     43   
j = -12    23    
j = -14   34     
j = -16   2      
j = -18 13       
j = -20          
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, 20]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 20]]
Out[4]=   
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[17, 1, 18, 4], X[5, 12, 6, 13], 
 
>   X[3, 8, 4, 9], X[13, 22, 14, 5], X[21, 14, 22, 15], X[9, 18, 10, 19], 
 
>   X[11, 20, 12, 21], X[19, 10, 20, 11], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, 11, -2, -3, 8, 
 
>    -10, 9, -7, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(21/2)    -(19/2)     3       5       6       7      6      6      3      2
q        - q        + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ----
                       17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
                      q       q       q       q       q      q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -32    -30    2     3     -22    -20    3     2     4     -12    2    2    2
-q    - q    - --- - --- - q    + q    + --- + --- + --- + q    + --- + -- + --
                28    26                  18    16    14           10    8    4
               q     q                   q     q     q            q     q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 20]][a, z]
Out[8]=   
   3     5      7      9
  a     a    4 a    2 a       3        7        9        3  3      5  3
-(--) - -- + ---- - ---- - 4 a  z + 7 a  z - 3 a  z - 2 a  z  + 2 a  z  + 
  z     z     z      z
 
       7  3    9  3    5  5    7  5
>   4 a  z  - a  z  + a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 20]][a, z]
Out[9]=   
                                  3    5      7      9
 4      6      8    10      12   a    a    4 a    2 a       3      5
a  - 5 a  - 6 a  + a   + 2 a   - -- + -- + ---- + ---- + 5 a  z - a  z - 
                                 z    z     z      z
 
        7        9      11      4  2       6  2       8  2      10  2
>   13 a  z - 8 a  z - a   z - a  z  + 15 a  z  + 21 a  z  - 2 a   z  - 
 
       12  2      3  3      5  3       7  3       9  3    11  3       6  4
>   7 a   z  - 3 a  z  - 5 a  z  + 16 a  z  + 17 a  z  - a   z  - 17 a  z  - 
 
        8  4    10  4      12  4      5  5       7  5       9  5      11  5
>   21 a  z  + a   z  + 5 a   z  + 4 a  z  - 10 a  z  - 11 a  z  + 3 a   z  - 
 
     4  6      6  6       8  6      10  6    12  6      5  7      7  7
>   a  z  + 7 a  z  + 11 a  z  + 2 a   z  - a   z  - 2 a  z  + 3 a  z  + 
 
       9  7    11  7      6  8      8  8    10  8    7  9    9  9
>   4 a  z  - a   z  - 2 a  z  - 3 a  z  - a   z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    2      1        1        3        2        3        4        2
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    22  9    18  8    18  7    16  6    14  6    14  5    12  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        4        3        3       3       3      3
>   ------ + ------ + ------ + ----- + ----- + ----- + ----
     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


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