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L10a117

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Acknowledgement

L10a117 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-27/2 + 2q-25/2 - 4q-23/2 + 5q-21/2 - 6q-19/2 + 6q-17/2 - 5q-15/2 + 4q-13/2 - 3q-11/2 + q-9/2 - q-7/2

A2 (sl(3)) Invariant: q-40 + q-36 + q-34 + q-32 + 2q-30 - q-28 + q-26 - q-24 + q-20 + 2q-16 + q-12

HOMFLY-PT Polynomial: - 6a7z - 11a7z3 - 6a7z5 - a7z7 - a9z-1 - 3a9z - 7a9z3 - 5a9z5 - a9z7 + a11z-1 + 4a11z + 4a11z3 + a11z5

Kauffman Polynomial: 6a7z - 11a7z3 + 6a7z5 - a7z7 - a8z2 - 3a8z4 + 4a8z6 - a8z8 + a9z-1 - 4a9z + 5a9z3 - 5a9z5 + 4a9z7 - a9z9 - a10 + 3a10z2 - 10a10z4 + 11a10z6 - 3a10z8 + a11z-1 - 5a11z + 7a11z3 - 2a11z5 + 2a11z7 - a11z9 + 2a12z2 - 2a12z4 + 4a12z6 - 2a12z8 + 2a13z - 5a13z3 + 6a13z5 - 3a13z7 - a14z2 + 3a14z4 - 3a14z6 - 2a15z + 3a15z3 - 3a15z5 + a16z2 - 2a16z4 + a17z - a17z3

Khovanov Homology:
trqj r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -6          1
j = -8         11
j = -10        2  
j = -12       21  
j = -14      32   
j = -16     32    
j = -18    33     
j = -20   23      
j = -22  23       
j = -24  2        
j = -2612         
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[10, Alternating, 117]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 117]]
Out[4]=   
PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[4, 11, 5, 12], 
 
>   X[20, 9, 11, 10], X[16, 5, 17, 6], X[18, 7, 19, 8], X[6, 17, 7, 18], 
 
>   X[8, 19, 9, 20], X[10, 15, 1, 16]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -4, 6, -8, 7, -9, 5, -10}, 
 
>   {4, -1, 2, -3, 10, -6, 8, -7, 9, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(27/2)     2       4       5       6       6       5       4       3
-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
 
     -(9/2)    -(7/2)
>   q       - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -40    -36    -34    -32    2     -28    -26    -24    -20    2     -12
q    + q    + q    + q    + --- - q    + q    - q    + q    + --- + q
                             30                                16
                            q                                 q
In[8]:=
HOMFLYPT[Link[10, Alternating, 117]][a, z]
Out[8]=   
   9     11
  a     a        7        9        11         7  3      9  3      11  3
-(--) + --- - 6 a  z - 3 a  z + 4 a   z - 11 a  z  - 7 a  z  + 4 a   z  - 
  z      z
 
       7  5      9  5    11  5    7  7    9  7
>   6 a  z  - 5 a  z  + a   z  - a  z  - a  z
In[9]:=
Kauffman[Link[10, Alternating, 117]][a, z]
Out[9]=   
        9    11
  10   a    a        7        9        11        13        15      17
-a   + -- + --- + 6 a  z - 4 a  z - 5 a   z + 2 a   z - 2 a   z + a   z - 
       z     z
 
     8  2      10  2      12  2    14  2    16  2       7  3      9  3
>   a  z  + 3 a   z  + 2 a   z  - a   z  + a   z  - 11 a  z  + 5 a  z  + 
 
       11  3      13  3      15  3    17  3      8  4       10  4      12  4
>   7 a   z  - 5 a   z  + 3 a   z  - a   z  - 3 a  z  - 10 a   z  - 2 a   z  + 
 
       14  4      16  4      7  5      9  5      11  5      13  5      15  5
>   3 a   z  - 2 a   z  + 6 a  z  - 5 a  z  - 2 a   z  + 6 a   z  - 3 a   z  + 
 
       8  6       10  6      12  6      14  6    7  7      9  7      11  7
>   4 a  z  + 11 a   z  + 4 a   z  - 3 a   z  - a  z  + 4 a  z  + 2 a   z  - 
 
       13  7    8  8      10  8      12  8    9  9    11  9
>   3 a   z  - a  z  - 3 a   z  - 2 a   z  - a  z  - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -8    -6      1         1        2        2        2        3        2
q   + q   + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 
             28  10    26  10    26  9    24  8    22  8    22  7    20  7
            q   t     q   t     q   t    q   t    q   t    q   t    q   t
 
      3        3        3        3        2        3        2        2
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     20  6    18  6    18  5    16  5    16  4    14  4    14  3    12  3
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      1        2       1
>   ------ + ------ + ----
     12  2    10  2    8
    q   t    q   t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a117
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