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L10a98

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Acknowledgement

L10a98 as Morse Link
DrawMorseLink

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

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

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

A2 (sl(3)) Invariant: q-40 + q-30 - q-28 + q-26 + q-22 + 2q-20 + q-18 + 2q-16 + q-12

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

Kauffman Polynomial: - a7z-1 + 7a7z - 11a7z3 + 6a7z5 - a7z7 + a8 - 3a8z2 - 2a8z4 + 4a8z6 - a8z8 - a9z-1 + 2a9z - a9z3 - 3a9z5 + 4a9z7 - a9z9 + 5a10z2 - 15a10z4 + 13a10z6 - 3a10z8 - 3a11z + 5a11z3 - 3a11z5 + 3a11z7 - a11z9 + 5a12z2 - 9a12z4 + 7a12z6 - 2a12z8 + a13z - 3a13z3 + 4a13z5 - 2a13z7 - a14z2 + 2a14z4 - 2a14z6 + a15z3 - 2a15z5 + 2a16z2 - 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       11  
j = -14      32   
j = -16     21    
j = -18    33     
j = -20   12      
j = -22  23       
j = -24 12        
j = -26 1         
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, 98]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 98]]
Out[4]=   
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], 
 
>   X[18, 7, 19, 8], X[20, 15, 9, 16], X[16, 19, 17, 20], X[8, 9, 1, 10], 
 
>   X[4, 13, 5, 14], X[6, 17, 7, 18]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -9, 4, -10, 5, -8}, 
 
>   {8, -1, 2, -3, 9, -4, 6, -7, 10, -5, 7, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(27/2)     2       3       4       5       5       4       3       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    -30    -28    -26    -22    2     -18    2     -12
q    + q    - q    + q    + q    + --- + q    + --- + q
                                    20           16
                                   q            q
In[8]:=
HOMFLYPT[Link[10, Alternating, 98]][a, z]
Out[8]=   
   7     9
  a     a       7        11         7  3      9  3      11  3      7  5
-(--) + -- - 7 a  z + 3 a   z - 11 a  z  - 6 a  z  + 4 a   z  - 6 a  z  - 
  z     z
 
       9  5    11  5    7  7    9  7
>   5 a  z  + a   z  - a  z  - a  z
In[9]:=
Kauffman[Link[10, Alternating, 98]][a, z]
Out[9]=   
      7    9
 8   a    a       7        9        11      13      17        8  2      10  2
a  - -- - -- + 7 a  z + 2 a  z - 3 a   z + a   z + a   z - 3 a  z  + 5 a   z  + 
     z    z
 
       12  2    14  2      16  2       7  3    9  3      11  3      13  3
>   5 a   z  - a   z  + 2 a   z  - 11 a  z  - a  z  + 5 a   z  - 3 a   z  + 
 
     15  3    17  3      8  4       10  4      12  4      14  4      16  4
>   a   z  - a   z  - 2 a  z  - 15 a   z  - 9 a   z  + 2 a   z  - 2 a   z  + 
 
       7  5      9  5      11  5      13  5      15  5      8  6       10  6
>   6 a  z  - 3 a  z  - 3 a   z  + 4 a   z  - 2 a   z  + 4 a  z  + 13 a   z  + 
 
       12  6      14  6    7  7      9  7      11  7      13  7    8  8
>   7 a   z  - 2 a   z  - a  z  + 4 a  z  + 3 a   z  - 2 a   z  - a  z  - 
 
       10  8      12  8    9  9    11  9
>   3 a   z  - 2 a   z  - a  z  - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -8    -6      1        1        1        2        2        3        1
q   + q   + ------- + ------ + ------ + ------ + ------ + ------ + ------ + 
             28  10    26  9    24  9    24  8    22  8    22  7    20  7
            q   t     q   t    q   t    q   t    q   t    q   t    q   t
 
      2        3        3        2        1        3        2        1
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     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 L10a98
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