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L11a152

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Acknowledgement

L11a152 as Morse Link
DrawMorseLink

PD Presentation: X8192 X10,3,11,4 X12,17,13,18 X14,5,15,6 X4,13,5,14 X18,11,19,12 X22,19,7,20 X20,15,21,16 X16,21,17,22 X2738 X6,9,1,10

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

Jones Polynomial: q-27/2 - 3q-25/2 + 7q-23/2 - 12q-21/2 + 16q-19/2 - 19q-17/2 + 19q-15/2 - 17q-13/2 + 12q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

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

HOMFLY-PT Polynomial: - a5z - 2a5z3 - a5z5 - a7z-1 - 6a7z - 8a7z3 - 3a7z5 - 3a9z3 - 2a9z5 + 2a11z-1 + 5a11z + 3a11z3 - a13z-1 - a13z

Kauffman Polynomial: - a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 - a7z-1 + 7a7z - 11a7z3 + 11a7z5 - 6a7z7 + a8 - 4a8z2 + 3a8z4 + 5a8z6 - 6a8z8 + 3a9z - 11a9z3 + 14a9z5 - 3a9z7 - 4a9z9 - 3a10 + 13a10z2 - 30a10z4 + 32a10z6 - 13a10z8 - a10z10 + 2a11z-1 - 6a11z + a11z3 + 10a11z7 - 8a11z9 - 5a12 + 25a12z2 - 44a12z4 + 38a12z6 - 12a12z8 - a12z10 + a13z-1 + a13z - 7a13z3 + 6a13z5 + 4a13z7 - 4a13z9 - 2a14 + 7a14z2 - 12a14z4 + 13a14z6 - 5a14z8 + 2a15z - 6a15z3 + 8a15z5 - 3a15z7 - 2a16z2 + 3a16z4 - a16z6

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4           1
j = -6          31
j = -8         5  
j = -10        73  
j = -12       105   
j = -14      97    
j = -16     1010     
j = -18    710      
j = -20   59       
j = -22  27        
j = -24 15         
j = -26 2          
j = -281           


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, 152]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 152]]
Out[4]=   
PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[12, 17, 13, 18], X[14, 5, 15, 6], 
 
>   X[4, 13, 5, 14], X[18, 11, 19, 12], X[22, 19, 7, 20], X[20, 15, 21, 16], 
 
>   X[16, 21, 17, 22], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -5, 4, -11}, 
 
>   {10, -1, 11, -2, 6, -3, 5, -4, 8, -9, 3, -6, 7, -8, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(27/2)     3       7      12      16      19      19      17      12
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 
            25/2    23/2    21/2    19/2    17/2    15/2    13/2    11/2
           q       q       q       q       q       q       q       q
 
     8      3      -(5/2)
>   ---- + ---- - q
     9/2    7/2
    q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    -40    -38    3     3     2     4     -24    3     2     5     -16
-q    - q    + q    - --- + --- - --- + --- + q    + --- - --- + --- - q    + 
                       36    32    30    28           22    20    18
                      q     q     q     q            q     q     q
 
     3     2     -8
>   --- - --- + q
     12    10
    q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 152]][a, z]
Out[8]=   
   7       11    13
  a     2 a     a      5        7        11      13        5  3      7  3
-(--) + ----- - --- - a  z - 6 a  z + 5 a   z - a   z - 2 a  z  - 8 a  z  - 
  z       z      z
 
       9  3      11  3    5  5      7  5      9  5
>   3 a  z  + 3 a   z  - a  z  - 3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 152]][a, z]
Out[9]=   
                              7      11    13
 8      10      12      14   a    2 a     a      5        7        9
a  - 3 a   - 5 a   - 2 a   - -- + ----- + --- - a  z + 7 a  z + 3 a  z - 
                             z      z      z
 
       11      13        15      6  2      8  2       10  2       12  2
>   6 a   z + a   z + 2 a   z - a  z  - 4 a  z  + 13 a   z  + 25 a   z  + 
 
       14  2      16  2      5  3       7  3       9  3    11  3      13  3
>   7 a   z  - 2 a   z  + 2 a  z  - 11 a  z  - 11 a  z  + a   z  - 7 a   z  - 
 
       15  3      6  4      8  4       10  4       12  4       14  4
>   6 a   z  + 4 a  z  + 3 a  z  - 30 a   z  - 44 a   z  - 12 a   z  + 
 
       16  4    5  5       7  5       9  5      13  5      15  5      6  6
>   3 a   z  - a  z  + 11 a  z  + 14 a  z  + 6 a   z  + 8 a   z  - 3 a  z  + 
 
       8  6       10  6       12  6       14  6    16  6      7  7      9  7
>   5 a  z  + 32 a   z  + 38 a   z  + 13 a   z  - a   z  - 6 a  z  - 3 a  z  + 
 
        11  7      13  7      15  7      8  8       10  8       12  8
>   10 a   z  + 4 a   z  - 3 a   z  - 6 a  z  - 13 a   z  - 12 a   z  - 
 
       14  8      9  9      11  9      13  9    10  10    12  10
>   5 a   z  - 4 a  z  - 8 a   z  - 4 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         2         1        5        2        7        5
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             28  11    26  10    24  10    24  9    22  9    22  8    20  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      9        7        10       10       10       9        7        10
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     20  7    18  7    18  6    16  6    16  5    14  5    14  4    12  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      5        7        3        5      3
>   ------ + ------ + ------ + ----- + ----
     12  3    10  3    10  2    8  2    6
    q   t    q   t    q   t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a152
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