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L10n13

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Acknowledgement

L10n13 as Morse Link
DrawMorseLink

PD Presentation: X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X17,2,18,3

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

Jones Polynomial: - q-17/2 + 2q-15/2 - 3q-13/2 + 3q-11/2 - 3q-9/2 + 3q-7/2 - 3q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant: q-30 + q-26 - q-18 + q-16 - q-14 + q-12 + q-10 + 2q-8 + 2q-6 + q-4 + q-2

HOMFLY-PT Polynomial: - 2a3z-1 - 7a3z - 5a3z3 - a3z5 + 4a5z-1 + 11a5z + 12a5z3 + 6a5z5 + a5z7 - 3a7z-1 - 7a7z - 5a7z3 - a7z5 + a9z-1 + a9z

Kauffman Polynomial: - 2a3z-1 + 9a3z - 12a3z3 + 6a3z5 - a3z7 + 2a4 - 4a4z2 - 2a4z4 + 4a4z6 - a4z8 - 4a5z-1 + 18a5z - 30a5z3 + 20a5z5 - 4a5z7 + 3a6 - 12a6z2 + 9a6z4 + a6z6 - a6z8 - 3a7z-1 + 11a7z - 17a7z3 + 13a7z5 - 3a7z7 + 3a8 - 10a8z2 + 11a8z4 - 3a8z6 - a9z-1 + a9z + a9z3 - a9z5 + a10 - 2a10z2 - a11z

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 0        1
j = -2         
j = -4      31 
j = -6     11  
j = -8    22   
j = -10  121    
j = -12  22     
j = -14 12      
j = -16 1       
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, NonAlternating, 13]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 13]]
Out[4]=   
PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[9, 14, 10, 15], 
 
>   X[3, 8, 4, 9], X[5, 13, 6, 12], X[13, 5, 14, 20], X[11, 16, 12, 17], 
 
>   X[15, 10, 16, 11], X[17, 2, 18, 3]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 9, -8, 6, -7, 4, -9, 8, -10, -2, 
 
>    3, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(17/2)     2       3       3      3      3      3      -(3/2)      1
-q        + ----- - ----- + ----- - ---- + ---- - ---- + q       - -------
             15/2    13/2    11/2    9/2    7/2    5/2             Sqrt[q]
            q       q       q       q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -30    -26    -18    -16    -14    -12    -10   2    2     -4    -2
q    + q    - q    + q    - q    + q    + q    + -- + -- + q   + q
                                                  8    6
                                                 q    q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 13]][a, z]
Out[8]=   
    3      5      7    9
-2 a    4 a    3 a    a       3         5        7      9        3  3
----- + ---- - ---- + -- - 7 a  z + 11 a  z - 7 a  z + a  z - 5 a  z  + 
  z      z      z     z
 
        5  3      7  3    3  5      5  5    7  5    5  7
>   12 a  z  - 5 a  z  - a  z  + 6 a  z  - a  z  + a  z
In[9]:=
Kauffman[Link[10, NonAlternating, 13]][a, z]
Out[9]=   
                              3      5      7    9
   4      6      8    10   2 a    4 a    3 a    a       3         5
2 a  + 3 a  + 3 a  + a   - ---- - ---- - ---- - -- + 9 a  z + 18 a  z + 
                            z      z      z     z
 
        7      9      11        4  2       6  2       8  2      10  2
>   11 a  z + a  z - a   z - 4 a  z  - 12 a  z  - 10 a  z  - 2 a   z  - 
 
        3  3       5  3       7  3    9  3      4  4      6  4       8  4
>   12 a  z  - 30 a  z  - 17 a  z  + a  z  - 2 a  z  + 9 a  z  + 11 a  z  + 
 
       3  5       5  5       7  5    9  5      4  6    6  6      8  6    3  7
>   6 a  z  + 20 a  z  + 13 a  z  - a  z  + 4 a  z  + a  z  - 3 a  z  - a  z  - 
 
       5  7      7  7    4  8    6  8
>   4 a  z  - 3 a  z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6   3      1        1        1        2        2        1        2
q   + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
       4    18  6    16  5    14  5    14  4    12  4    10  4    12  3
      q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      2        1        2      2      1     t     2
>   ------ + ------ + ----- + ---- + ---- + -- + t
     10  3    10  2    8  2    8      6      4
    q   t    q   t    q  t    q  t   q  t   q


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