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L10a60

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Acknowledgement

L10a60 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-9/2 - 3q-7/2 + 6q-5/2 - 9q-3/2 + 11q-1/2 - 13q1/2 + 11q3/2 - 10q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant: - q-14 - 2q-8 + 2q-6 - q-4 + q-2 + 3 + 5q4 + 3q8 + q10 - 2q12 + q14 - q16

HOMFLY-PT Polynomial: 3a-3z + 3a-3z3 + a-3z5 - 2a-1z-1 - 8a-1z - 10a-1z3 - 5a-1z5 - a-1z7 + 3az-1 + 8az + 7az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial: a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 - 2a-4z2 + 5a-4z4 - 5a-4z6 + 3a-3z - 7a-3z3 + 8a-3z5 - 6a-3z7 - 6a-2z2 + 8a-2z4 - 4a-2z8 - 2a-1z-1 + 14a-1z - 30a-1z3 + 31a-1z5 - 11a-1z7 - a-1z9 + 3 - 10z2 + 6z4 + 10z6 - 7z8 - 3az-1 + 13az - 28az3 + 29az5 - 8az7 - az9 + 3a2 - 10a2z2 + 7a2z4 + 4a2z6 - 3a2z8 - a3z-1 + 3a3z - 8a3z3 + 9a3z5 - 3a3z7 + a4 - 3a4z2 + 3a4z4 - a4z6

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 12          1
j = 10         2 
j = 8        41 
j = 6       62  
j = 4      65   
j = 2     75    
j = 0    57     
j = -2   46      
j = -4  25       
j = -6 14        
j = -8 2         
j = -101          


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, 60]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 60]]
Out[4]=   
PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[14, 6, 15, 5], 
 
>   X[18, 13, 19, 14], X[16, 19, 17, 20], X[12, 17, 13, 18], X[2, 7, 3, 8], 
 
>   X[4, 12, 5, 11], X[6, 16, 1, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -8, 2, -9, 4, -10}, 
 
>   {8, -1, 3, -2, 9, -7, 5, -4, 10, -6, 7, -5, 6, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(9/2)    3      6      9       11                       3/2       5/2
q       - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q    - 10 q    + 
           7/2    5/2    3/2   Sqrt[q]
          q      q      q
 
       7/2      9/2    11/2
>   6 q    - 3 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -14   2    2     -4    -2      4      8    10      12    14    16
3 - q    - -- + -- - q   + q   + 5 q  + 3 q  + q   - 2 q   + q   - q
            8    6
           q    q
In[8]:=
HOMFLYPT[Link[10, Alternating, 60]][a, z]
Out[8]=   
             3                                   3       3
-2    3 a   a    3 z   8 z              3     3 z    10 z         3    3  3
--- + --- - -- + --- - --- + 8 a z - 2 a  z + ---- - ----- + 7 a z  - a  z  + 
a z    z    z     3     a                       3      a
                 a                             a
 
     5      5             7
    z    5 z         5   z
>   -- - ---- + 2 a z  - --
     3    a              a
    a
In[9]:=
Kauffman[Link[10, Alternating, 60]][a, z]
Out[9]=   
                             3
       2    4    2    3 a   a    z    3 z   14 z               3         2
3 + 3 a  + a  - --- - --- - -- - -- + --- + ---- + 13 a z + 3 a  z - 10 z  + 
                a z    z    z     5    3     a
                                 a    a
 
     2      2      2                           3      3       3
    z    2 z    6 z        2  2      4  2   3 z    7 z    30 z          3
>   -- - ---- - ---- - 10 a  z  - 3 a  z  + ---- - ---- - ----- - 28 a z  - 
     6     4      2                           5      3      a
    a     a      a                           a      a
 
                      4      4      4                          5      5
       3  3      4   z    5 z    8 z       2  4      4  4   3 z    8 z
>   8 a  z  + 6 z  - -- + ---- + ---- + 7 a  z  + 3 a  z  - ---- + ---- + 
                      6     4      2                          5      3
                     a     a      a                          a      a
 
        5                                  6                        7       7
    31 z          5      3  5       6   5 z       2  6    4  6   6 z    11 z
>   ----- + 29 a z  + 9 a  z  + 10 z  - ---- + 4 a  z  - a  z  - ---- - ----- - 
      a                                   4                        3      a
                                         a                        a
 
                                 8              9
         7      3  7      8   4 z       2  8   z       9
>   8 a z  - 3 a  z  - 7 z  - ---- - 3 a  z  - -- - a z
                                2              a
                               a
In[10]:=
Kh[L][q, t]
Out[10]=   
       2     1        2       1       4       2       5       4     5    6
7 + 7 q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 
            10  5    8  4    6  4    6  3    4  3    4  2    2  2   t    2
           q   t    q  t    q  t    q  t    q  t    q  t    q  t        q  t
 
       2        4        4  2      6  2      6  3      8  3    8  4
>   5 q  t + 6 q  t + 5 q  t  + 6 q  t  + 2 q  t  + 4 q  t  + q  t  + 
 
       10  4    12  5
>   2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a60
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