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L11a224

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Acknowledgement

L11a224 as Morse Link
DrawMorseLink

PD Presentation: X8192 X16,6,17,5 X18,10,19,9 X10,20,11,19 X2,11,3,12 X12,3,13,4 X4758 X20,14,21,13 X22,16,7,15 X6,18,1,17 X14,22,15,21

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

Jones Polynomial: - q-5/2 + 3q-3/2 - 7q-1/2 + 10q1/2 - 15q3/2 + 16q5/2 - 17q7/2 + 15q9/2 - 11q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant: q-6 - q-4 + 3q-2 + 3q2 + 4q4 + 6q8 - 3q10 + 2q12 - 2q14 - 2q16 + q18 - 3q20 + q22 - q24

HOMFLY-PT Polynomial: 2a-5z-1 + 7a-5z + 9a-5z3 + 5a-5z5 + a-5z7 - 5a-3z-1 - 16a-3z - 25a-3z3 - 19a-3z5 - 7a-3z7 - a-3z9 + 3a-1z-1 + 8a-1z + 9a-1z3 + 5a-1z5 + a-1z7

Kauffman Polynomial: a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + a-8 - 5a-8z2 + 7a-8z4 - 6a-8z6 - 7a-7z3 + 11a-7z5 - 8a-7z7 - 6a-6z4 + 12a-6z6 - 8a-6z8 + 2a-5z-1 - 9a-5z + 15a-5z3 - 10a-5z5 + 10a-5z7 - 6a-5z9 - 5a-4 + 18a-4z2 - 27a-4z4 + 26a-4z6 - 6a-4z8 - 2a-4z10 + 5a-3z-1 - 20a-3z + 37a-3z3 - 40a-3z5 + 32a-3z7 - 10a-3z9 - 5a-2 + 14a-2z2 - 24a-2z4 + 19a-2z6 - a-2z8 - 2a-2z10 + 3a-1z-1 - 9a-1z + 8a-1z3 - 12a-1z5 + 13a-1z7 - 4a-1z9 + 2z2 - 11z4 + 11z6 - 3z8 + 2az - 5az3 + 4az5 - az7

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 18           1
j = 16          2 
j = 14         51 
j = 12        62  
j = 10       95   
j = 8      86    
j = 6     89     
j = 4    78      
j = 2   49       
j = 0  36        
j = -2 15         
j = -4 2          
j = -61           


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, 224]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 224]]
Out[4]=   
PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[18, 10, 19, 9], X[10, 20, 11, 19], 
 
>   X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 7, 5, 8], X[20, 14, 21, 13], 
 
>   X[22, 16, 7, 15], X[6, 18, 1, 17], X[14, 22, 15, 21]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -5, 6, -7, 2, -10}, 
 
>   {7, -1, 3, -4, 5, -6, 8, -11, 9, -2, 10, -3, 4, -8, 11, -9}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(5/2)    3        7                       3/2       5/2       7/2
-q       + ---- - ------- + 10 Sqrt[q] - 15 q    + 16 q    - 17 q    + 
            3/2   Sqrt[q]
           q
 
        9/2       11/2      13/2      15/2    17/2
>   15 q    - 11 q     + 7 q     - 3 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -6    -4   3       2      4      8      10      12      14      16    18
q   - q   + -- + 3 q  + 4 q  + 6 q  - 3 q   + 2 q   - 2 q   - 2 q   + q   - 
             2
            q
 
       20    22    24
>   3 q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 224]][a, z]
Out[8]=   
                                          3       3      3      5       5
 2      5      3    7 z   16 z   8 z   9 z    25 z    9 z    5 z    19 z
---- - ---- + --- + --- - ---- + --- + ---- - ----- + ---- + ---- - ----- + 
 5      3     a z    5      3     a      5      3      a       5      3
a  z   a  z         a      a            a      a              a      a
 
       5    7      7    7    9
    5 z    z    7 z    z    z
>   ---- + -- - ---- + -- - --
     a      5     3    a     3
           a     a          a
In[9]:=
Kauffman[Link[11, Alternating, 224]][a, z]
Out[9]=   
                                                                       2
 -8   5    5     2      5      3    9 z   20 z   9 z              2   z
a   - -- - -- + ---- + ---- + --- - --- - ---- - --- + 2 a z + 2 z  + --- - 
       4    2    5      3     a z    5      3     a                    10
      a    a    a  z   a  z         a      a                          a
 
       2       2       2      3      3       3       3      3
    5 z    18 z    14 z    2 z    7 z    15 z    37 z    8 z         3
>   ---- + ----- + ----- + ---- - ---- + ----- + ----- + ---- - 5 a z  - 
      8      4       2       9      7      5       3      a
     a      a       a       a      a      a       a
 
             4       4      4       4       4      5       5       5       5
        4   z     7 z    6 z    27 z    24 z    3 z    11 z    10 z    40 z
>   11 z  - --- + ---- - ---- - ----- - ----- - ---- + ----- - ----- - ----- - 
             10     8      6      4       2       9      7       5       3
            a      a      a      a       a       a      a       a       a
 
        5                       6       6       6       6      7       7
    12 z         5       6   6 z    12 z    26 z    19 z    8 z    10 z
>   ----- + 4 a z  + 11 z  - ---- + ----- + ----- + ----- - ---- + ----- + 
      a                        8      6       4       2       7      5
                              a      a       a       a       a      a
 
        7       7                    8      8    8      9       9      9
    32 z    13 z       7      8   8 z    6 z    z    6 z    10 z    4 z
>   ----- + ----- - a z  - 3 z  - ---- - ---- - -- - ---- - ----- - ---- - 
      3       a                     6      4     2     5      3      a
     a                             a      a     a     a      a
 
       10      10
    2 z     2 z
>   ----- - -----
      4       2
     a       a
In[10]:=
Kh[L][q, t]
Out[10]=   
                                                          2
   2      4     1       2       1     3      5     6   4 q       4        6
9 q  + 7 q  + ----- + ----- + ----- + -- + ----- + - + ---- + 8 q  t + 8 q  t + 
               6  4    4  3    2  3    2    2  2   t    t
              q  t    q  t    q  t    t    q  t
 
       6  2      8  2      8  3      10  3      10  4      12  4      12  5
>   9 q  t  + 8 q  t  + 6 q  t  + 9 q   t  + 5 q   t  + 6 q   t  + 2 q   t  + 
 
       14  5    14  6      16  6    18  7
>   5 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a224
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