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L11a95

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Acknowledgement

L11a95 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-19/2 + 3q-17/2 - 6q-15/2 + 12q-13/2 - 16q-11/2 + 19q-9/2 - 20q-7/2 + 16q-5/2 - 14q-3/2 + 8q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant: q-30 + q-28 - q-26 + q-24 - 2q-22 - 6q-20 - 4q-16 + 2q-14 + 4q-12 + 2q-10 + 8q-8 - q-6 + 4q-4 + q-2 - 2 + 2q2 - q4

HOMFLY-PT Polynomial: az + 2az3 + az5 - 4a3z-1 - 9a3z - 8a3z3 - 4a3z5 - a3z7 + 8a5z-1 + 14a5z + 10a5z3 + 3a5z5 - 5a7z-1 - 7a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial: - z2 + 2z4 - z6 + 2az - 8az3 + 10az5 - 4az7 - 2a2z2 - 3a2z4 + 12a2z6 - 6a2z8 - 4a3z-1 + 14a3z - 25a3z3 + 24a3z5 - 2a3z7 - 4a3z9 + 8a4 - 13a4z2 - 5a4z4 + 26a4z6 - 12a4z8 - a4z10 - 8a5z-1 + 21a5z - 33a5z3 + 27a5z5 - 7a5z9 + 14a6 - 29a6z2 + 12a6z4 + 13a6z6 - 10a6z8 - a6z10 - 5a7z-1 + 9a7z - 16a7z3 + 17a7z5 - 6a7z7 - 3a7z9 + 9a8 - 22a8z2 + 18a8z4 - 3a8z6 - 4a8z8 - a9z-1 - a9z + 2a9z3 + 3a9z5 - 4a9z7 + 2a10 - 5a10z2 + 6a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 4           1
j = 2          3 
j = 0         51 
j = -2        93  
j = -4       97   
j = -6      117    
j = -8     89     
j = -10    811      
j = -12   48       
j = -14  28        
j = -16 14         
j = -18 2          
j = -201           


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, 95]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 95]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 12, 15, 11], X[18, 15, 19, 16], 
 
>   X[16, 9, 17, 10], X[10, 17, 11, 18], X[22, 19, 5, 20], X[20, 7, 21, 8], 
 
>   X[8, 21, 9, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 8, -9, 5, -6, 3, -2, 11, -3, 4, -5, 6, -4, 
 
>    7, -8, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(19/2)     3       6      12      16      19     20     16     14
-q        + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 
             17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
            q       q       q       q       q      q      q      q
 
       8                   3/2
>   ------- - 4 Sqrt[q] + q
    Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -30    -28    -26    -24    2     6     4     2     4     2    8     -6
-2 + q    + q    - q    + q    - --- - --- - --- + --- + --- + --- + -- - q   + 
                                  22    20    16    14    12    10    8
                                 q     q     q     q     q     q     q
 
    4     -2      2    4
>   -- + q   + 2 q  - q
     4
    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 95]][a, z]
Out[8]=   
    3      5      7    9
-4 a    8 a    5 a    a             3         5        7      9          3
----- + ---- - ---- + -- + a z - 9 a  z + 14 a  z - 7 a  z + a  z + 2 a z  - 
  z      z      z     z
 
       3  3       5  3      7  3      5      3  5      5  5    3  7
>   8 a  z  + 10 a  z  - 3 a  z  + a z  - 4 a  z  + 3 a  z  - a  z
In[9]:=
Kauffman[Link[11, Alternating, 95]][a, z]
Out[9]=   
                                 3      5      7    9
   4       6      8      10   4 a    8 a    5 a    a                3
8 a  + 14 a  + 9 a  + 2 a   - ---- - ---- - ---- - -- + 2 a z + 14 a  z + 
                               z      z      z     z
 
        5        7      9      11      2      2  2       4  2       6  2
>   21 a  z + 9 a  z - a  z - a   z - z  - 2 a  z  - 13 a  z  - 29 a  z  - 
 
        8  2      10  2        3       3  3       5  3       7  3      9  3
>   22 a  z  - 5 a   z  - 8 a z  - 25 a  z  - 33 a  z  - 16 a  z  + 2 a  z  + 
 
       11  3      4      2  4      4  4       6  4       8  4      10  4
>   2 a   z  + 2 z  - 3 a  z  - 5 a  z  + 12 a  z  + 18 a  z  + 6 a   z  + 
 
          5       3  5       5  5       7  5      9  5    11  5    6
>   10 a z  + 24 a  z  + 27 a  z  + 17 a  z  + 3 a  z  - a   z  - z  + 
 
        2  6       4  6       6  6      8  6      10  6        7      3  7
>   12 a  z  + 26 a  z  + 13 a  z  - 3 a  z  - 3 a   z  - 4 a z  - 2 a  z  - 
 
       7  7      9  7      2  8       4  8       6  8      8  8      3  9
>   6 a  z  - 4 a  z  - 6 a  z  - 12 a  z  - 10 a  z  - 4 a  z  - 4 a  z  - 
 
       5  9      7  9    4  10    6  10
>   7 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
7    9      1        2        1        4        2        8        4
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    20  8    18  7    16  7    16  6    14  6    14  5    12  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      8        8        11       8       9      11      7      9
>   ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 t + 
     12  4    10  4    10  3    8  3    8  2    6  2    6      4
    q   t    q   t    q   t    q  t    q  t    q  t    q  t   q  t
 
    3 t    2      2  2    4  3
>   --- + t  + 3 q  t  + q  t
     2
    q


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