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L11n164

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Acknowledgement

L11n164 as Morse Link
DrawMorseLink

PD Presentation: X8192 X10,3,11,4 X12,17,13,18 X14,5,15,6 X4,13,5,14 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10

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

Jones Polynomial: - q-17/2 + 2q-15/2 - 4q-13/2 + 5q-11/2 - 6q-9/2 + 6q-7/2 - 6q-5/2 + 4q-3/2 - 3q-1/2 + q1/2

A2 (sl(3)) Invariant: q-26 + q-22 + 2q-20 + 2q-16 + q-10 + 2q-6 + 1 - q2

HOMFLY-PT Polynomial: az + az3 - 3a3z - 3a3z3 - a3z5 - a5z-1 - 3a5z - 3a5z3 - a5z5 + a7z-1 + 2a7z + a7z3

Kauffman Polynomial: - z2 + az - 3az3 - a2z2 - a2z6 + 3a3z - 4a3z3 + 4a3z5 - 2a3z7 + 4a4z6 - 2a4z8 + a5z-1 - 2a5z - 3a5z3 + 7a5z5 - a5z9 - a6 + 3a6z2 - 11a6z4 + 14a6z6 - 4a6z8 + a7z-1 - 10a7z3 + 8a7z5 + a7z7 - a7z9 + 3a8z2 - 11a8z4 + 9a8z6 - 2a8z8 + 4a9z - 8a9z3 + 5a9z5 - a9z7

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 2         1
j = 0        2 
j = -2       32 
j = -4      31  
j = -6     33   
j = -8    33    
j = -10   23     
j = -12  23      
j = -14 13       
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, NonAlternating, 164]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 164]]
Out[4]=   
PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[12, 17, 13, 18], X[14, 5, 15, 6], 
 
>   X[4, 13, 5, 14], X[18, 11, 19, 12], X[19, 7, 20, 22], X[15, 21, 16, 20], 
 
>   X[21, 17, 22, 16], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -5, 4, -11}, 
 
>   {10, -1, 11, -2, 6, -3, 5, -4, -8, 9, 3, -6, -7, 8, -9, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(17/2)     2       4       5      6      6      6      4        3
-q        + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 
             15/2    13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q       q       q      q      q      q
 
>   Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -26    -22    2     2     -10   2     2
1 + q    + q    + --- + --- + q    + -- - q
                   20    16           6
                  q     q            q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 164]][a, z]
Out[8]=   
   5     7
  a     a             3        5        7        3      3  3      5  3
-(--) + -- + a z - 3 a  z - 3 a  z + 2 a  z + a z  - 3 a  z  - 3 a  z  + 
  z     z
 
     7  3    3  5    5  5
>   a  z  - a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 164]][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 - z  - a  z  + 3 a  z  + 
      z    z
 
       8  2        3      3  3      5  3       7  3      9  3       6  4
>   3 a  z  - 3 a z  - 4 a  z  - 3 a  z  - 10 a  z  - 8 a  z  - 11 a  z  - 
 
        8  4      3  5      5  5      7  5      9  5    2  6      4  6
>   11 a  z  + 4 a  z  + 7 a  z  + 8 a  z  + 5 a  z  - a  z  + 4 a  z  + 
 
        6  6      8  6      3  7    7  7    9  7      4  8      6  8
>   14 a  z  + 9 a  z  - 2 a  z  + a  z  - a  z  - 2 a  z  - 4 a  z  - 
 
       8  8    5  9    7  9
>   2 a  z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    2      1        1        1        3        2        3        2
2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     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
 
      3        3       3       3       3       3      1      3      2
>   ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q  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


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