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L11a292

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Acknowledgement

L11a292 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-11/2 - 3q-9/2 + 7q-7/2 - 12q-5/2 + 14q-3/2 - 17q-1/2 + 16q1/2 - 14q3/2 + 10q5/2 - 6q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant: - q-16 + q-14 - 2q-12 + 3q-8 - q-6 + 5q-4 + 2 + 2q2 - 2q4 + 3q6 - 2q8 + q12 - q14 + q16

HOMFLY-PT Polynomial: - 2a-3z - 3a-3z3 - a-3z5 - a-1z-1 + a-1z + 5a-1z3 + 4a-1z5 + a-1z7 + az-1 + 4az + 6az3 + 4az5 + az7 - 3a3z - 3a3z3 - a3z5

Kauffman Polynomial: - 4a-5z3 + 4a-5z5 - a-5z7 + 4a-4z2 - 14a-4z4 + 12a-4z6 - 3a-4z8 - 4a-3z + 12a-3z3 - 19a-3z5 + 15a-3z7 - 4a-3z9 + 3a-2z2 - 10a-2z4 + 10a-2z6 + a-2z8 - 2a-2z10 - a-1z-1 - 2a-1z + 17a-1z3 - 28a-1z5 + 27a-1z7 - 9a-1z9 + 1 - z2 - 5z4 + 15z6 - 4z8 - 2z10 - az-1 + 8az - 17az3 + 14az5 + 2az7 - 5az9 - 4a2z2 - 2a2z4 + 11a2z6 - 8a2z8 + 6a3z - 16a3z3 + 16a3z5 - 9a3z7 - 3a4z2 + 6a4z4 - 6a4z6 + 2a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 12           1
j = 10          2 
j = 8         41 
j = 6        62  
j = 4       84   
j = 2      86    
j = 0     98     
j = -2    710      
j = -4   57       
j = -6  27        
j = -8 15         
j = -10 2          
j = -121           


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, 292]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 292]]
Out[4]=   
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 10, 19, 9], 
 
>   X[22, 18, 9, 17], X[8, 21, 1, 22], X[20, 13, 21, 14], X[14, 8, 15, 7], 
 
>   X[16, 6, 17, 5], X[6, 16, 7, 15], X[4, 20, 5, 19]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -11, 9, -10, 8, -6}, 
 
>   {4, -1, 2, -3, 7, -8, 10, -9, 5, -4, 11, -7, 6, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(11/2)    3      7      12     14      17                       3/2
q        - ---- + ---- - ---- + ---- - ------- + 16 Sqrt[q] - 14 q    + 
            9/2    7/2    5/2    3/2   Sqrt[q]
           q      q      q      q
 
        5/2      7/2      9/2    11/2
>   10 q    - 6 q    + 3 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -16    -14    2    3     -6   5       2      4      6      8    12    14
2 - q    + q    - --- + -- - q   + -- + 2 q  - 2 q  + 3 q  - 2 q  + q   - q   + 
                   12    8          4
                  q     q          q
 
     16
>   q
In[8]:=
HOMFLYPT[Link[11, Alternating, 292]][a, z]
Out[8]=   
                                           3      3                       5
   1     a   2 z   z              3     3 z    5 z         3      3  3   z
-(---) + - - --- + - + 4 a z - 3 a  z - ---- + ---- + 6 a z  - 3 a  z  - -- + 
  a z    z    3    a                      3     a                         3
             a                           a                               a
 
       5                     7
    4 z         5    3  5   z       7
>   ---- + 4 a z  - a  z  + -- + a z
     a                      a
In[9]:=
Kauffman[Link[11, Alternating, 292]][a, z]
Out[9]=   
                                                   2      2
     1    a   4 z   2 z              3      2   4 z    3 z       2  2
1 - --- - - - --- - --- + 8 a z + 6 a  z - z  + ---- + ---- - 4 a  z  - 
    a z   z    3     a                            4      2
              a                                  a      a
 
                         3       3       3
       4  2    6  2   4 z    12 z    17 z          3       3  3      5  3
>   3 a  z  + a  z  - ---- + ----- + ----- - 17 a z  - 16 a  z  + 2 a  z  - 
                        5      3       a
                       a      a
 
               4       4                                  5       5       5
       4   14 z    10 z       2  4      4  4    6  4   4 z    19 z    28 z
>   5 z  - ----- - ----- - 2 a  z  + 6 a  z  - a  z  + ---- - ----- - ----- + 
             4       2                                   5      3       a
            a       a                                   a      a
 
                                               6       6
          5       3  5      5  5       6   12 z    10 z        2  6      4  6
>   14 a z  + 16 a  z  - 3 a  z  + 15 z  + ----- + ----- + 11 a  z  - 6 a  z  - 
                                             4       2
                                            a       a
 
     7       7       7                                8    8                9
    z    15 z    27 z         7      3  7      8   3 z    z       2  8   4 z
>   -- + ----- + ----- + 2 a z  - 9 a  z  - 4 z  - ---- + -- - 8 a  z  - ---- - 
     5     3       a                                 4     2               3
    a     a                                         a     a               a
 
       9                       10
    9 z         9      10   2 z
>   ---- - 5 a z  - 2 z   - -----
     a                        2
                             a
In[10]:=
Kh[L][q, t]
Out[10]=   
    10     1        2        1       5       2       7       5      7
9 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 
     2    12  5    10  4    8  4    8  3    6  3    6  2    4  2    4
    q    q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
     7              2        2  2      4  2      4  3      6  3      6  4
>   ---- + 8 t + 8 q  t + 6 q  t  + 8 q  t  + 4 q  t  + 6 q  t  + 2 q  t  + 
     2
    q  t
 
       8  4    8  5      10  5    12  6
>   4 q  t  + q  t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a292
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