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L10n11

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Acknowledgement

L10n11 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X2,14,3,13

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

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

A2 (sl(3)) Invariant: q-20 + q-16 + q-14 - 2q-12 - q-8 + q-4 + 3 + q2 + 2q4 + 2q6

HOMFLY-PT Polynomial: - 2a-1z-1 - 2a-1z + 4az-1 + 7az + 3az3 - 3a3z-1 - 7a3z - 4a3z3 - a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial: - 2a-1z-1 + 5a-1z - 3a-1z3 + 2 - z6 - 4az-1 + 15az - 17az3 + 7az5 - 2az7 + 3a2 - 5a2z2 + a2z6 - a2z8 - 3a3z-1 + 13a3z - 20a3z3 + 13a3z5 - 4a3z7 + 3a4 - 9a4z2 + 6a4z4 - a4z8 - a5z-1 + 2a5z - 3a5z3 + 5a5z5 - 2a5z7 + a6 - 4a6z2 + 6a6z4 - 2a6z6 - a7z + 3a7z3 - a7z5

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 4        2
j = 2       1 
j = 0      42 
j = -2     33  
j = -4    32   
j = -6   23    
j = -8  13     
j = -10 12      
j = -12 1       
j = -141        


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


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