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L10a106

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Acknowledgement

L10a106 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-17/2 - 3q-15/2 + 6q-13/2 - 11q-11/2 + 13q-9/2 - 14q-7/2 + 13q-5/2 - 11q-3/2 + 7q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant: - q-26 + q-22 - q-20 + 4q-18 + q-14 + 2q-12 - 2q-10 + 4q-8 - 2q-6 + 2q-4 + q-2 - 1 + 2q2 - q4

HOMFLY-PT Polynomial: 2az3 + az5 - a3z-1 - 5a3z - 6a3z3 - 4a3z5 - a3z7 + a5z-1 + 5a5z + 6a5z3 + 2a5z5 - 2a7z - a7z3

Kauffman Polynomial: - z2 + 2z4 - z6 + az - 8az3 + 11az5 - 4az7 - a2z2 - 3a2z4 + 11a2z6 - 5a2z8 - a3z-1 + 9a3z - 27a3z3 + 31a3z5 - 7a3z7 - 2a3z9 + a4 + a4z2 - 12a4z4 + 24a4z6 - 11a4z8 - a5z-1 + 10a5z - 30a5z3 + 33a5z5 - 10a5z7 - 2a5z9 - a6z2 - 2a6z4 + 7a6z6 - 6a6z8 + a7z - 8a7z3 + 10a7z5 - 7a7z7 - a8z2 + 4a8z4 - 5a8z6 - a9z + 3a9z3 - 3a9z5 + a10z2 - a10z4

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 4          1
j = 2         3 
j = 0        41 
j = -2       73  
j = -4      75   
j = -6     76    
j = -8    67     
j = -10   57      
j = -12  27       
j = -14 14        
j = -16 2         
j = -181          


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, 106]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 106]]
Out[4]=   
PD[X[10, 1, 11, 2], X[18, 11, 19, 12], X[20, 5, 9, 6], X[14, 7, 15, 8], 
 
>   X[12, 4, 13, 3], X[16, 14, 17, 13], X[6, 15, 7, 16], X[8, 9, 1, 10], 
 
>   X[4, 19, 5, 20], X[2, 18, 3, 17]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 5, -9, 3, -7, 4, -8}, 
 
>   {8, -1, 2, -5, 6, -4, 7, -6, 10, -2, 9, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(17/2)     3       6      11      13     14     13     11       7
q        - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 
            15/2    13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q       q      q      q      q
 
                 3/2
>   4 Sqrt[q] + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -26    -22    -20    4     -14    2     2    4    2    2     -2      2
-1 - q    + q    - q    + --- + q    + --- - --- + -- - -- + -- + q   + 2 q  - 
                           18           12    10    8    6    4
                          q            q     q     q    q    q
 
     4
>   q
In[8]:=
HOMFLYPT[Link[10, Alternating, 106]][a, z]
Out[8]=   
   3     5
  a     a       3        5        7          3      3  3      5  3    7  3
-(--) + -- - 5 a  z + 5 a  z - 2 a  z + 2 a z  - 6 a  z  + 6 a  z  - a  z  + 
  z     z
 
       5      3  5      5  5    3  7
>   a z  - 4 a  z  + 2 a  z  - a  z
In[9]:=
Kauffman[Link[10, Alternating, 106]][a, z]
Out[9]=   
      3    5
 4   a    a             3         5      7      9      2    2  2    4  2
a  - -- - -- + a z + 9 a  z + 10 a  z + a  z - a  z - z  - a  z  + a  z  - 
     z    z
 
     6  2    8  2    10  2        3       3  3       5  3      7  3      9  3
>   a  z  - a  z  + a   z  - 8 a z  - 27 a  z  - 30 a  z  - 8 a  z  + 3 a  z  + 
 
       4      2  4       4  4      6  4      8  4    10  4         5
>   2 z  - 3 a  z  - 12 a  z  - 2 a  z  + 4 a  z  - a   z  + 11 a z  + 
 
        3  5       5  5       7  5      9  5    6       2  6       4  6
>   31 a  z  + 33 a  z  + 10 a  z  - 3 a  z  - z  + 11 a  z  + 24 a  z  + 
 
       6  6      8  6        7      3  7       5  7      7  7      2  8
>   7 a  z  - 5 a  z  - 4 a z  - 7 a  z  - 10 a  z  - 7 a  z  - 5 a  z  - 
 
        4  8      6  8      3  9      5  9
>   11 a  z  - 6 a  z  - 2 a  z  - 2 a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
5    7      1        2        1        4        2        7        5
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    18  7    16  6    14  6    14  5    12  5    12  4    10  4
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      7        6       7       7      6      7           3 t    2      2  2
>   ------ + ----- + ----- + ----- + ---- + ---- + 4 t + --- + t  + 3 q  t  + 
     10  3    8  3    8  2    6  2    6      4            2
    q   t    q  t    q  t    q  t    q  t   q  t         q
 
     4  3
>   q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a106
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L10a107