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L11a85

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Acknowledgement

L11a85 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X14,10,15,9 X10,14,11,13 X20,15,21,16 X18,7,19,8 X8,19,9,20 X22,17,5,18 X16,21,17,22 X2536 X4,11,1,12

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

Jones Polynomial: q-23/2 - 2q-21/2 + 5q-19/2 - 8q-17/2 + 11q-15/2 - 14q-13/2 + 12q-11/2 - 12q-9/2 + 9q-7/2 - 6q-5/2 + 3q-3/2 - q-1/2

A2 (sl(3)) Invariant: - q-36 - 2q-34 - q-32 - 2q-30 - q-28 + 2q-26 + 5q-22 + 3q-20 + 3q-18 + 4q-16 - 2q-14 + 2q-12 - 2q-10 + q-6 - q-4 + q-2

HOMFLY-PT Polynomial: - 2a3z - 3a3z3 - a3z5 + 3a5z + 5a5z3 + 4a5z5 + a5z7 - 3a7z-1 - 11a7z - 11a7z3 - 3a7z5 + 5a9z-1 + 9a9z + 3a9z3 - 2a11z-1 - a11z

Kauffman Polynomial: 2a3z - 5a3z3 + 4a3z5 - a3z7 + 5a4z2 - 14a4z4 + 12a4z6 - 3a4z8 + a5z - 6a5z3 + 8a5z7 - 3a5z9 + 10a6z2 - 29a6z4 + 27a6z6 - 5a6z8 - a6z10 - 3a7z-1 + 8a7z - 13a7z3 + a7z5 + 14a7z7 - 6a7z9 + 5a8 - 3a8z2 - 15a8z4 + 22a8z6 - 6a8z8 - a8z10 - 5a9z-1 + 14a9z - 20a9z3 + 12a9z5 + a9z7 - 3a9z9 + 5a10 - 10a10z2 + 3a10z4 + 4a10z6 - 4a10z8 - 2a11z-1 + 5a11z - 6a11z3 + 5a11z5 - 4a11z7 + 2a12z4 - 3a12z6 + 2a13z3 - 2a13z5 - a14 + 2a14z2 - a14z4

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 0           1
j = -2          2 
j = -4         41 
j = -6        63  
j = -8       63   
j = -10      66    
j = -12     86     
j = -14    47      
j = -16   47       
j = -18  14        
j = -20 14         
j = -22 1          
j = -241           


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, 85]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 85]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 10, 15, 9], X[10, 14, 11, 13], 
 
>   X[20, 15, 21, 16], X[18, 7, 19, 8], X[8, 19, 9, 20], X[22, 17, 5, 18], 
 
>   X[16, 21, 17, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 6, -7, 3, -4, 11, -2, 4, -3, 5, -9, 8, -6, 
 
>    7, -5, 9, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(23/2)     2       5       8      11      14      12      12     9      6
q        - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 
            21/2    19/2    17/2    15/2    13/2    11/2    9/2    7/2    5/2
           q       q       q       q       q       q       q      q      q
 
     3        1
>   ---- - -------
     3/2   Sqrt[q]
    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -36    2     -32    2     -28    2     5     3     3     4     2     2
-q    - --- - q    - --- - q    + --- + --- + --- + --- + --- - --- + --- - 
         34           30           26    22    20    18    16    14    12
        q            q            q     q     q     q     q     q     q
 
     2     -6    -4    -2
>   --- + q   - q   + q
     10
    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 85]][a, z]
Out[8]=   
    7      9      11
-3 a    5 a    2 a        3        5         7        9      11        3  3
----- + ---- - ----- - 2 a  z + 3 a  z - 11 a  z + 9 a  z - a   z - 3 a  z  + 
  z      z       z
 
       5  3       7  3      9  3    3  5      5  5      7  5    5  7
>   5 a  z  - 11 a  z  + 3 a  z  - a  z  + 4 a  z  - 3 a  z  + a  z
In[9]:=
Kauffman[Link[11, Alternating, 85]][a, z]
Out[9]=   
                        7      9      11
   8      10    14   3 a    5 a    2 a        3      5        7         9
5 a  + 5 a   - a   - ---- - ---- - ----- + 2 a  z + a  z + 8 a  z + 14 a  z + 
                      z      z       z
 
       11        4  2       6  2      8  2       10  2      14  2      3  3
>   5 a   z + 5 a  z  + 10 a  z  - 3 a  z  - 10 a   z  + 2 a   z  - 5 a  z  - 
 
       5  3       7  3       9  3      11  3      13  3       4  4       6  4
>   6 a  z  - 13 a  z  - 20 a  z  - 6 a   z  + 2 a   z  - 14 a  z  - 29 a  z  - 
 
        8  4      10  4      12  4    14  4      3  5    7  5       9  5
>   15 a  z  + 3 a   z  + 2 a   z  - a   z  + 4 a  z  + a  z  + 12 a  z  + 
 
       11  5      13  5       4  6       6  6       8  6      10  6
>   5 a   z  - 2 a   z  + 12 a  z  + 27 a  z  + 22 a  z  + 4 a   z  - 
 
       12  6    3  7      5  7       7  7    9  7      11  7      4  8
>   3 a   z  - a  z  + 8 a  z  + 14 a  z  + a  z  - 4 a   z  - 3 a  z  - 
 
       6  8      8  8      10  8      5  9      7  9      9  9    6  10    8  10
>   5 a  z  - 6 a  z  - 4 a   z  - 3 a  z  - 6 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3    4      1        1        1        4        1        4        4
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 6    4    24  9    22  8    20  8    20  7    18  7    18  6    16  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      7        4        7        8        6        6        6        6
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 
     16  5    14  5    14  4    12  4    12  3    10  3    10  2    8  2
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q  t
 
     3      6     t    2 t    2
>   ---- + ---- + -- + --- + t
     8      6      4    2
    q  t   q  t   q    q


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