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L10a14

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Acknowledgement

L10a14 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,4,11,3 X14,8,15,7 X20,16,5,15 X16,9,17,10 X8,19,9,20 X18,13,19,14 X12,17,13,18 X2536 X4,12,1,11

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

Jones Polynomial: - q-13/2 + 3q-11/2 - 7q-9/2 + 12q-7/2 - 15q-5/2 + 16q-3/2 - 16q-1/2 + 12q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant: q-20 + 3q-14 - 3q-12 + q-10 - q-8 - 3q-6 + 3q-4 - 2q-2 + 6 + q2 + q4 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial: - 2a-1z-1 - 2a-1z - 2a-1z3 - a-1z5 + 4az-1 + 7az + 7az3 + 4az5 + az7 - 3a3z-1 - 7a3z - 6a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial: a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 - 2a-1z-1 + 5a-1z - 10a-1z3 + 15a-1z5 - 8a-1z7 + 2 - 2z2 + z4 + 8z6 - 7z8 - 4az-1 + 15az - 33az3 + 38az5 - 14az7 - 2az9 + 3a2 - 5a2z2 - 4a2z4 + 18a2z6 - 12a2z8 - 3a3z-1 + 13a3z - 28a3z3 + 30a3z5 - 11a3z7 - 2a3z9 + 3a4 - 7a4z2 + 5a4z4 + 3a4z6 - 5a4z8 - a5z-1 + 2a5z - 4a5z3 + 7a5z5 - 5a5z7 + a6 - 3a6z2 + 5a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8          1
j = 6         3 
j = 4        61 
j = 2       63  
j = 0      106   
j = -2     88    
j = -4    78     
j = -6   58      
j = -8  27       
j = -10 15        
j = -12 2         
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, Alternating, 14]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 14]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[14, 8, 15, 7], X[20, 16, 5, 15], 
 
>   X[16, 9, 17, 10], X[8, 19, 9, 20], X[18, 13, 19, 14], X[12, 17, 13, 18], 
 
>   X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 2, -10}, {9, -1, 3, -6, 5, -2, 10, -8, 7, -3, 4, -5, 8, -7, 
 
>    6, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(13/2)     3      7      12     15     16      16                      3/2
-q        + ----- - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 9 q    + 
             11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q      q      q      q
 
       5/2    7/2
>   4 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -20    3     3     -10    -8   3    3    2     2    4      6      8    10
6 + q    + --- - --- + q    - q   - -- + -- - -- + q  + q  + 3 q  - 2 q  + q
            14    12                 6    4    2
           q     q                  q    q    q
In[8]:=
HOMFLYPT[Link[10, Alternating, 14]][a, z]
Out[8]=   
               3    5                                      3
-2    4 a   3 a    a    2 z              3        5     2 z         3
--- + --- - ---- + -- - --- + 7 a z - 7 a  z + 2 a  z - ---- + 7 a z  - 
a z    z     z     z     a                               a
 
                       5
       3  3    5  3   z         5      3  5      7
>   6 a  z  + a  z  - -- + 4 a z  - 2 a  z  + a z
                      a
In[9]:=
Kauffman[Link[10, Alternating, 14]][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
 
                            2                                  3       3
       5      7        2   z       2  2      4  2      6  2   z    10 z
>   2 a  z - a  z - 2 z  - -- - 5 a  z  - 7 a  z  - 3 a  z  + -- - ----- - 
                            2                                  3     a
                           a                                  a
 
                                                     4
          3       3  3      5  3      7  3    4   5 z       2  4      4  4
>   33 a z  - 28 a  z  - 4 a  z  + 2 a  z  + z  + ---- - 4 a  z  + 5 a  z  + 
                                                    2
                                                   a
 
               5       5                                                    6
       6  4   z    15 z          5       3  5      5  5    7  5      6   4 z
>   5 a  z  - -- + ----- + 38 a z  + 30 a  z  + 7 a  z  - a  z  + 8 z  - ---- + 
               3     a                                                     2
              a                                                           a
 
                                      7
        2  6      4  6      6  6   8 z          7       3  7      5  7      8
>   18 a  z  + 3 a  z  - 3 a  z  - ---- - 14 a z  - 11 a  z  - 5 a  z  - 7 z  - 
                                    a
 
        2  8      4  8        9      3  9
>   12 a  z  - 5 a  z  - 2 a z  - 2 a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     8      1        2        1        5        2       7       5       8
10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
      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
 
      7      8      8              2        2  2      4  2    4  3      6  3
>   ----- + ---- + ---- + 6 t + 6 q  t + 3 q  t  + 6 q  t  + q  t  + 3 q  t  + 
     4  2    4      2
    q  t    q  t   q  t
 
     8  4
>   q  t


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