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L11a276

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Acknowledgement

L11a276 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-21/2 + 2q-19/2 - 4q-17/2 + 5q-15/2 - 6q-13/2 + 7q-11/2 - 7q-9/2 + 5q-7/2 - 5q-5/2 + 3q-3/2 - 2q-1/2 + q1/2

A2 (sl(3)) Invariant: q-30 + 2q-26 + q-24 + q-22 - q-18 + 2q-16 + 3q-12 + q-10 + q-8 - q-4 - 1

HOMFLY-PT Polynomial: a3z-1 + 7a3z + 11a3z3 + 6a3z5 + a3z7 - 3a5z-1 - 21a5z - 34a5z3 - 24a5z5 - 8a5z7 - a5z9 + 2a7z-1 + 10a7z + 12a7z3 + 6a7z5 + a7z7

Kauffman Polynomial: - a2 + 7a2z2 - 11a2z4 + 6a2z6 - a2z8 + a3z-1 - 7a3z + 18a3z3 - 23a3z5 + 12a3z7 - 2a3z9 - 3a4 + 20a4z2 - 30a4z4 + 11a4z6 + 2a4z8 - a4z10 + 3a5z-1 - 23a5z + 48a5z3 - 53a5z5 + 28a5z7 - 5a5z9 - 3a6 + 19a6z2 - 30a6z4 + 16a6z6 - a6z10 + 2a7z-1 - 12a7z + 24a7z3 - 22a7z5 + 13a7z7 - 3a7z9 + 4a8z2 - 6a8z4 + 8a8z6 - 3a8z8 + a9z - 2a9z3 + 5a9z5 - 3a9z7 - a10z2 + 3a10z4 - 3a10z6 - 2a11z + 3a11z3 - 3a11z5 + a12z2 - 2a12z4 + a13z - a13z3

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 2           1
j = 0          1 
j = -2         21 
j = -4        31  
j = -6       33   
j = -8      42    
j = -10     33     
j = -12    34      
j = -14   23       
j = -16  23        
j = -18  2         
j = -2012          
j = -221           


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, 276]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 276]]
Out[4]=   
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], 
 
>   X[18, 7, 19, 8], X[22, 20, 9, 19], X[20, 16, 21, 15], X[16, 22, 17, 21], 
 
>   X[8, 9, 1, 10], X[4, 13, 5, 14], X[6, 17, 7, 18]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -10, 4, -11, 5, -9}, 
 
>   {9, -1, 2, -3, 10, -4, 7, -8, 11, -5, 6, -7, 8, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(21/2)     2       4       5       6       7      7      5      5      3
-q        + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - 
             19/2    17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
            q       q       q       q       q       q      q      q      q
 
       2
>   ------- + Sqrt[q]
    Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -30    2     -24    -22    -18    2     3     -10    -8    -4
-1 + q    + --- + q    + q    - q    + --- + --- + q    + q   - q
             26                         16    12
            q                          q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 276]][a, z]
Out[8]=   
 3      5      7
a    3 a    2 a       3         5         7         3  3       5  3
-- - ---- + ---- + 7 a  z - 21 a  z + 10 a  z + 11 a  z  - 34 a  z  + 
z     z      z
 
        7  3      3  5       5  5      7  5    3  7      5  7    7  7    5  9
>   12 a  z  + 6 a  z  - 24 a  z  + 6 a  z  + a  z  - 8 a  z  + a  z  - a  z
In[9]:=
Kauffman[Link[11, Alternating, 276]][a, z]
Out[9]=   
                     3      5      7
  2      4      6   a    3 a    2 a       3         5         7      9
-a  - 3 a  - 3 a  + -- + ---- + ---- - 7 a  z - 23 a  z - 12 a  z + a  z - 
                    z     z      z
 
       11      13        2  2       4  2       6  2      8  2    10  2
>   2 a   z + a   z + 7 a  z  + 20 a  z  + 19 a  z  + 4 a  z  - a   z  + 
 
     12  2       3  3       5  3       7  3      9  3      11  3    13  3
>   a   z  + 18 a  z  + 48 a  z  + 24 a  z  - 2 a  z  + 3 a   z  - a   z  - 
 
        2  4       4  4       6  4      8  4      10  4      12  4       3  5
>   11 a  z  - 30 a  z  - 30 a  z  - 6 a  z  + 3 a   z  - 2 a   z  - 23 a  z  - 
 
        5  5       7  5      9  5      11  5      2  6       4  6       6  6
>   53 a  z  - 22 a  z  + 5 a  z  - 3 a   z  + 6 a  z  + 11 a  z  + 16 a  z  + 
 
       8  6      10  6       3  7       5  7       7  7      9  7    2  8
>   8 a  z  - 3 a   z  + 12 a  z  + 28 a  z  + 13 a  z  - 3 a  z  - a  z  + 
 
       4  8      8  8      3  9      5  9      7  9    4  10    6  10
>   2 a  z  - 3 a  z  - 2 a  z  - 5 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3    3      1        1        2        2        2        3        2
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 6    4    22  8    20  8    20  7    18  6    16  6    16  5    14  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        3        4        3        3        4      2      3     t
>   ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + 
     14  4    12  4    12  3    10  3    10  2    8  2    8      6      4
    q   t    q   t    q   t    q   t    q   t    q  t    q  t   q  t   q
 
                2
    2 t    2   t     2  3
>   --- + t  + -- + q  t
     2          2
    q          q


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