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L11a370

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Acknowledgement

L11a370 as Morse Link
DrawMorseLink

PD Presentation: X12,1,13,2 X20,8,21,7 X14,3,15,4 X6,15,7,16 X16,5,17,6 X4,17,5,18 X22,20,11,19 X18,9,19,10 X2,11,3,12 X10,13,1,14 X8,22,9,21

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

Jones Polynomial: - q-17/2 + 2q-15/2 - 5q-13/2 + 9q-11/2 - 13q-9/2 + 15q-7/2 - 17q-5/2 + 15q-3/2 - 12q-1/2 + 8q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant: q-26 + q-22 + 3q-20 - 2q-18 + 2q-16 + 2q-14 - 2q-12 + 3q-10 - q-8 + 2q-6 + q-4 - 2q-2 + 3 - 3q2 + 2q6 - q8

HOMFLY-PT Polynomial: a-1z3 + az - az5 - 3a3z - 4a3z3 - 2a3z5 - a5z-1 - 3a5z - 2a5z3 - a5z5 + a7z-1 + 2a7z + a7z3

Kauffman Polynomial: - a-2z4 + 2a-1z3 - 4a-1z5 - 2z2 + 8z4 - 8z6 + az - 6az3 + 13az5 - 10az7 - 3a2z2 + 4a2z4 + 7a2z6 - 8a2z8 + 3a3z - 13a3z3 + 20a3z5 - 4a3z7 - 4a3z9 - 9a4z4 + 22a4z6 - 9a4z8 - a4z10 + a5z-1 - 2a5z - 4a5z3 + a5z5 + 11a5z7 - 6a5z9 - a6 + 5a6z2 - 14a6z4 + 15a6z6 - 3a6z8 - a6z10 + a7z-1 - 7a7z3 + 3a7z5 + 4a7z7 - 2a7z9 + 4a8z2 - 10a8z4 + 8a8z6 - 2a8z8 + 4a9z - 8a9z3 + 5a9z5 - a9z7

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 6           1
j = 4          3 
j = 2         51 
j = 0        73  
j = -2       96   
j = -4      86    
j = -6     79     
j = -8    68      
j = -10   37       
j = -12  26        
j = -14 14         
j = -16 1          
j = -181           


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, Alternating, 370]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 370]]
Out[4]=   
PD[X[12, 1, 13, 2], X[20, 8, 21, 7], X[14, 3, 15, 4], X[6, 15, 7, 16], 
 
>   X[16, 5, 17, 6], X[4, 17, 5, 18], X[22, 20, 11, 19], X[18, 9, 19, 10], 
 
>   X[2, 11, 3, 12], X[10, 13, 1, 14], X[8, 22, 9, 21]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 3, -6, 5, -4, 2, -11, 8, -10}, 
 
>   {9, -1, 10, -3, 4, -5, 6, -8, 7, -2, 11, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(17/2)     2       5       9      13     15     17     15      12
-q        + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 
             15/2    13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q       q       q      q      q      q
 
                   3/2    5/2
>   8 Sqrt[q] - 4 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -26    -22    3     2     2     2     2     3     -8   2     -4   2
3 + q    + q    + --- - --- + --- + --- - --- + --- - q   + -- + q   - -- - 
                   20    18    16    14    12    10          6          2
                  q     q     q     q     q     q           q          q
 
       2      6    8
>   3 q  + 2 q  - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 370]][a, z]
Out[8]=   
   5     7                                     3
  a     a             3        5        7     z       3  3      5  3    7  3
-(--) + -- + a z - 3 a  z - 3 a  z + 2 a  z + -- - 4 a  z  - 2 a  z  + a  z  - 
  z     z                                     a
 
       5      3  5    5  5
>   a z  - 2 a  z  - a  z
In[9]:=
Kauffman[Link[11, Alternating, 370]][a, z]
Out[9]=   
       5    7
  6   a    a             3        5        9        2      2  2      6  2
-a  + -- + -- + a z + 3 a  z - 2 a  z + 4 a  z - 2 z  - 3 a  z  + 5 a  z  + 
      z    z
 
                 3
       8  2   2 z         3       3  3      5  3      7  3      9  3      4
>   4 a  z  + ---- - 6 a z  - 13 a  z  - 4 a  z  - 7 a  z  - 8 a  z  + 8 z  - 
               a
 
     4                                                5
    z       2  4      4  4       6  4       8  4   4 z          5       3  5
>   -- + 4 a  z  - 9 a  z  - 14 a  z  - 10 a  z  - ---- + 13 a z  + 20 a  z  + 
     2                                              a
    a
 
     5  5      7  5      9  5      6      2  6       4  6       6  6
>   a  z  + 3 a  z  + 5 a  z  - 8 z  + 7 a  z  + 22 a  z  + 15 a  z  + 
 
       8  6         7      3  7       5  7      7  7    9  7      2  8
>   8 a  z  - 10 a z  - 4 a  z  + 11 a  z  + 4 a  z  - a  z  - 8 a  z  - 
 
       4  8      6  8      8  8      3  9      5  9      7  9    4  10    6  10
>   9 a  z  - 3 a  z  - 2 a  z  - 4 a  z  - 6 a  z  - 2 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    6      1        1        1        4        2        6        3
7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     2    18  8    16  7    14  7    14  6    12  6    12  5    10  5
    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      7        6       8       7       9       8      6      9
>   ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 
     10  4    8  4    8  3    6  3    6  2    4  2    4      2
    q   t    q  t    q  t    q  t    q  t    q  t    q  t   q  t
 
       2      2  2      4  2    6  3
>   5 q  t + q  t  + 3 q  t  + q  t


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