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L11a76

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Acknowledgement

L11a76 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-25/2 - 2q-23/2 + 5q-21/2 - 7q-19/2 + 9q-17/2 - 12q-15/2 + 11q-13/2 - 11q-11/2 + 8q-9/2 - 6q-7/2 + 3q-5/2 - q-3/2

A2 (sl(3)) Invariant: - q-40 - 2q-38 - 2q-34 - 3q-32 + 2q-30 + 2q-26 + 5q-24 + 2q-22 + 3q-20 + q-16 + 2q-14 - 2q-12 + 2q-10 + q-8 - 2q-6 + q-4

HOMFLY-PT Polynomial: - a3z3 - 2a5z - 3a5z3 - 2a7z-1 - 4a7z - 4a7z3 + 2a9z-1 + a9z - 2a9z3 + a11z-1 + 3a11z - a13z-1

Kauffman Polynomial: - a3z3 - 3a4z4 - 2a5z + 5a5z3 - 6a5z5 - 2a6z2 + 10a6z4 - 8a6z6 - 2a7z-1 + 7a7z - 11a7z3 + 18a7z5 - 9a7z7 + 2a8 - 5a8z2 + 2a8z4 + 13a8z6 - 7a8z8 - 2a9z-1 + 12a9z - 26a9z3 + 19a9z5 + 4a9z7 - 4a9z9 - 4a10 + 19a10z2 - 46a10z4 + 38a10z6 - 7a10z8 - a10z10 + a11z-1 + 3a11z - 3a11z3 - 20a11z5 + 23a11z7 - 6a11z9 - 9a12 + 34a12z2 - 48a12z4 + 23a12z6 - a12z8 - a12z10 + a13z-1 + 6a13z3 - 15a13z5 + 10a13z7 - 2a13z9 - 4a14 + 12a14z2 - 13a14z4 + 6a14z6 - a14z8

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2           1
j = -4          31
j = -6         3  
j = -8        53  
j = -10       63   
j = -12      66    
j = -14     65     
j = -16    36      
j = -18   46       
j = -20  13        
j = -22 14         
j = -24 1          
j = -261           


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, 76]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 76]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 15, 21, 16], X[16, 7, 17, 8], 
 
>   X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[22, 13, 5, 14], 
 
>   X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, 8, -9, 3, -4, 7, -6, 
 
>    5, -3, 9, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(25/2)     2       5       7       9      12      11      11      8
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ---- - 
            23/2    21/2    19/2    17/2    15/2    13/2    11/2    9/2
           q       q       q       q       q       q       q       q
 
     6      3      -(3/2)
>   ---- + ---- - q
     7/2    5/2
    q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -40    2     2     3     2     2     5     2     3     -16    2     2
-q    - --- - --- - --- + --- + --- + --- + --- + --- + q    + --- - --- + 
         38    34    32    30    26    24    22    20           14    12
        q     q     q     q     q     q     q     q            q     q
 
     2     -8   2     -4
>   --- + q   - -- + q
     10          6
    q           q
In[8]:=
HOMFLYPT[Link[11, Alternating, 76]][a, z]
Out[8]=   
    7      9    11    13
-2 a    2 a    a     a        5        7      9        11      3  3      5  3
----- + ---- + --- - --- - 2 a  z - 4 a  z + a  z + 3 a   z - a  z  - 3 a  z  - 
  z      z      z     z
 
       7  3      9  3
>   4 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 76]][a, z]
Out[9]=   
                                  7      9    11    13
   8      10      12      14   2 a    2 a    a     a        5        7
2 a  - 4 a   - 9 a   - 4 a   - ---- - ---- + --- + --- - 2 a  z + 7 a  z + 
                                z      z      z     z
 
        9        11        6  2      8  2       10  2       12  2       14  2
>   12 a  z + 3 a   z - 2 a  z  - 5 a  z  + 19 a   z  + 34 a   z  + 12 a   z  - 
 
     3  3      5  3       7  3       9  3      11  3      13  3      4  4
>   a  z  + 5 a  z  - 11 a  z  - 26 a  z  - 3 a   z  + 6 a   z  - 3 a  z  + 
 
        6  4      8  4       10  4       12  4       14  4      5  5
>   10 a  z  + 2 a  z  - 46 a   z  - 48 a   z  - 13 a   z  - 6 a  z  + 
 
        7  5       9  5       11  5       13  5      6  6       8  6
>   18 a  z  + 19 a  z  - 20 a   z  - 15 a   z  - 8 a  z  + 13 a  z  + 
 
        10  6       12  6      14  6      7  7      9  7       11  7
>   38 a   z  + 23 a   z  + 6 a   z  - 9 a  z  + 4 a  z  + 23 a   z  + 
 
        13  7      8  8      10  8    12  8    14  8      9  9      11  9
>   10 a   z  - 7 a  z  - 7 a   z  - a   z  - a   z  - 4 a  z  - 6 a   z  - 
 
       13  9    10  10    12  10
>   2 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -4    -2      1         1         1        4        1        3        4
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             26  11    24  10    22  10    22  9    20  9    20  8    18  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      6        3        6        6        5        6        6        6
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     18  7    16  7    16  6    14  6    14  5    12  5    12  4    10  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        5       3       3      3
>   ------ + ----- + ----- + ----- + ----
     10  3    8  3    8  2    6  2    4
    q   t    q  t    q  t    q  t    q  t


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