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L11n324

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Acknowledgement

L11n324 as Morse Link
DrawMorseLink

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

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

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

A2 (sl(3)) Invariant: - q-24 - q-20 + 3q-16 + q-14 + 4q-12 + 3q-10 + 3q-8 + 4q-6 + q-4 + 4q-2 + 2 + 2q2 + 2q4

HOMFLY-PT Polynomial: z-2 + 4 + 2z2 - 2a2z-2 - 9a2 - 10a2z2 - 3a2z4 + a4z-2 + 7a4 + 10a4z2 + 5a4z4 + a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial: - z-2 + 6 - 8z2 + 3z4 + 2az-1 - 5az + az3 - az5 + az7 - 2a2z-2 + 13a2 - 31a2z2 + 25a2z4 - 9a2z6 + 2a2z8 + 2a3z-1 - 8a3z + 6a3z3 + 2a3z5 - 2a3z7 + a3z9 - a4z-2 + 9a4 - 26a4z2 + 32a4z4 - 16a4z6 + 4a4z8 - 3a5z + 8a5z3 - 2a5z5 - a5z7 + a5z9 + a6z2 + 4a6z4 - 5a6z6 + 2a6z8 + a7z - 4a7z5 + 2a7z7 - a8 + 4a8z2 - 6a8z4 + 2a8z6 + a9z - 3a9z3 + a9z5

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 3         2
j = 1        1 
j = -1       52 
j = -3      44  
j = -5     42   
j = -7    34    
j = -9   34     
j = -11  14      
j = -13 12       
j = -15 1        
j = -171         


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[11, NonAlternating, 324]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 324]]
Out[4]=   
PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[2, 16, 3, 15], 
 
>   X[16, 7, 17, 8], X[9, 18, 10, 19], X[11, 21, 12, 20], X[19, 22, 20, 13], 
 
>   X[13, 12, 14, 5], X[17, 1, 18, 4], X[21, 11, 22, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11, -7, 9}, 
 
>   {-9, 2, 4, -5, -10, 6, -8, 7, -11, 8}]
In[6]:=
Jones[L][q]
Out[6]=   
      -8   2    3    6    7    8    6    6
-3 - q   + -- - -- + -- - -- + -- - -- + - + 2 q
            7    6    5    4    3    2   q
           q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    -20    3     -14    4     3    3    4     -4   4       2      4
2 - q    - q    + --- + q    + --- + --- + -- + -- + q   + -- + 2 q  + 2 q
                   16           12    10    8    6          2
                  q            q     q     q    q          q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 324]][a, z]
Out[8]=   
                                  2    4
       2      4      6    -2   2 a    a       2       2  2       4  2
4 - 9 a  + 7 a  - 2 a  + z   - ---- + -- + 2 z  - 10 a  z  + 10 a  z  - 
                                 2     2
                                z     z
 
       6  2      2  4      4  4    6  4    4  6
>   3 a  z  - 3 a  z  + 5 a  z  - a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 324]][a, z]
Out[9]=   
                                 2    4            3
        2      4    8    -2   2 a    a    2 a   2 a               3
6 + 13 a  + 9 a  - a  - z   - ---- - -- + --- + ---- - 5 a z - 8 a  z - 
                                2     2    z     z
                               z     z
 
       5      7      9        2       2  2       4  2    6  2      8  2
>   3 a  z + a  z + a  z - 8 z  - 31 a  z  - 26 a  z  + a  z  + 4 a  z  + 
 
       3      3  3      5  3      9  3      4       2  4       4  4      6  4
>   a z  + 6 a  z  + 8 a  z  - 3 a  z  + 3 z  + 25 a  z  + 32 a  z  + 4 a  z  - 
 
       8  4      5      3  5      5  5      7  5    9  5      2  6       4  6
>   6 a  z  - a z  + 2 a  z  - 2 a  z  - 4 a  z  + a  z  - 9 a  z  - 16 a  z  - 
 
       6  6      8  6      7      3  7    5  7      7  7      2  8      4  8
>   5 a  z  + 2 a  z  + a z  - 2 a  z  - a  z  + 2 a  z  + 2 a  z  + 4 a  z  + 
 
       6  8    3  9    5  9
>   2 a  z  + a  z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
4    5     1        1        1        2        1        4        3       4
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 3   q    17  7    15  6    13  6    13  5    11  5    11  4    9  4    9  3
q        q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      3       4       4      2      4     2 t            3  2
>   ----- + ----- + ----- + ---- + ---- + --- + q t + 2 q  t
     7  3    7  2    5  2    5      3      q
    q  t    q  t    q  t    q  t   q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n324
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