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L11a388

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Acknowledgement

L11a388 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-13 + 2q-12 - 6q-11 + 10q-10 - 13q-9 + 16q-8 - 15q-7 + 16q-6 - 10q-5 + 7q-4 - 3q-3 + q-2

A2 (sl(3)) Invariant: - q-42 - 3q-40 - 2q-38 - 2q-36 - 5q-34 + 2q-32 + 3q-30 + 2q-28 + 8q-26 + 4q-24 + 7q-22 + 5q-20 + 3q-18 + 6q-16 - 2q-14 + q-12 + 2q-10 - 2q-8 + q-6

HOMFLY-PT Polynomial: a4z2 + a4z4 + a6z-2 + 4a6 + 5a6z2 + 3a6z4 - a8z-2 - a8 + 3a8z2 + 3a8z4 - 2a10z-2 - 7a10 - 6a10z2 + 3a12z-2 + 4a12 - a14z-2

Kauffman Polynomial: - a4z2 + a4z4 - 2a5z3 + 3a5z5 + a6z-2 - 4a6 + 8a6z2 - 8a6z4 + 6a6z6 - 2a7z-1 + 4a7z - a7z3 - 6a7z5 + 7a7z7 + a8z-2 - 3a8 + 9a8z2 - 4a8z4 - 7a8z6 + 7a8z8 + 2a9z-1 - 10a9z + 23a9z3 - 23a9z5 + 3a9z7 + 4a9z9 - 2a10z-2 + 4a10 - 5a10z2 + 17a10z4 - 28a10z6 + 10a10z8 + a10z10 + 10a11z-1 - 34a11z + 49a11z3 - 27a11z5 - 6a11z7 + 6a11z9 - 3a12z-2 + 5a12 - 7a12z2 + 19a12z4 - 22a12z6 + 5a12z8 + a12z10 + 8a13z-1 - 27a13z + 36a13z3 - 18a13z5 - a13z7 + 2a13z9 - a14z-2 + a14 - 2a14z2 + 7a14z4 - 7a14z6 + 2a14z8 + 2a15z-1 - 7a15z + 9a15z3 - 5a15z5 + a15z7

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -3           1
j = -5          31
j = -7         4  
j = -9        63  
j = -11       104   
j = -13      89    
j = -15     87     
j = -17    58      
j = -19   58       
j = -21  15        
j = -23 15         
j = -25 1          
j = -271           


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]=   
3
In[3]:=
Show[DrawMorseLink[Link[11, Alternating, 388]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 388]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], 
 
>   X[18, 11, 19, 12], X[22, 15, 9, 16], X[20, 17, 21, 18], X[16, 21, 17, 22], 
 
>   X[12, 19, 13, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, 
 
>   {11, -2, 5, -9, 4, -3, 6, -8, 7, -5, 9, -7, 8, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
  -13    2     6    10    13   16   15   16   10   7    3     -2
-q    + --- - --- + --- - -- + -- - -- + -- - -- + -- - -- + q
         12    11    10    9    8    7    6    5    4    3
        q     q     q     q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    3     2     2     5     2     3     2     8     4     7     5     3
-q    - --- - --- - --- - --- + --- + --- + --- + --- + --- + --- + --- + --- + 
         40    38    36    34    32    30    28    26    24    22    20    18
        q     q     q     q     q     q     q     q     q     q     q     q
 
     6     2     -12    2    2     -6
>   --- - --- + q    + --- - -- + q
     16    14           10    8
    q     q            q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 388]][a, z]
Out[8]=   
                             6    8      10      12    14
   6    8      10      12   a    a    2 a     3 a     a      4  2      6  2
4 a  - a  - 7 a   + 4 a   + -- - -- - ----- + ----- - --- + a  z  + 5 a  z  + 
                             2    2     2       2      2
                            z    z     z       z      z
 
       8  2      10  2    4  4      6  4      8  4
>   3 a  z  - 6 a   z  + a  z  + 3 a  z  + 3 a  z
In[9]:=
Kauffman[Link[11, Alternating, 388]][a, z]
Out[9]=   
                                      6    8      10      12    14      7
    6      8      10      12    14   a    a    2 a     3 a     a     2 a
-4 a  - 3 a  + 4 a   + 5 a   + a   + -- + -- - ----- - ----- - --- - ---- + 
                                      2    2     2       2      2     z
                                     z    z     z       z      z
 
       9       11      13      15
    2 a    10 a     8 a     2 a        7         9         11         13
>   ---- + ------ + ----- + ----- + 4 a  z - 10 a  z - 34 a   z - 27 a   z - 
     z       z        z       z
 
       15      4  2      6  2      8  2      10  2      12  2      14  2
>   7 a   z - a  z  + 8 a  z  + 9 a  z  - 5 a   z  - 7 a   z  - 2 a   z  - 
 
       5  3    7  3       9  3       11  3       13  3      15  3    4  4
>   2 a  z  - a  z  + 23 a  z  + 49 a   z  + 36 a   z  + 9 a   z  + a  z  - 
 
       6  4      8  4       10  4       12  4      14  4      5  5      7  5
>   8 a  z  - 4 a  z  + 17 a   z  + 19 a   z  + 7 a   z  + 3 a  z  - 6 a  z  - 
 
        9  5       11  5       13  5      15  5      6  6      8  6
>   23 a  z  - 27 a   z  - 18 a   z  - 5 a   z  + 6 a  z  - 7 a  z  - 
 
        10  6       12  6      14  6      7  7      9  7      11  7    13  7
>   28 a   z  - 22 a   z  - 7 a   z  + 7 a  z  + 3 a  z  - 6 a   z  - a   z  + 
 
     15  7      8  8       10  8      12  8      14  8      9  9      11  9
>   a   z  + 7 a  z  + 10 a   z  + 5 a   z  + 2 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]=   
 -5    -3      1         1         1        5        1        5        5
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             27  11    25  10    23  10    23  9    21  9    21  8    19  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      8        5        8        8        7        8        9        10
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     19  7    17  7    17  6    15  6    15  5    13  5    13  4    11  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      4        6       3       4      3
>   ------ + ----- + ----- + ----- + ----
     11  3    9  3    9  2    7  2    5
    q   t    q  t    q  t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a388
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