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L11n348

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Acknowledgement

L11n348 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,3,15,4 X11,20,12,21 X7,18,8,19 X17,22,18,13 X9,17,10,16 X15,11,16,10 X19,12,20,5 X21,8,22,9 X2536 X4,13,1,14

Gauss Code: {{1, -10, 2, -11}, {10, -1, -4, 9, -6, 7, -3, 8}, {11, -2, -7, 6, -5, 4, -8, 3, -9, 5}}

Jones Polynomial: - q-10 + 2q-9 - 2q-8 + 2q-7 + q-6 + 2q-4 - 2q-3 + 3q-2 - 2q-1 + 1

A2 (sl(3)) Invariant: - q-32 - q-30 + q-28 + 3q-24 + 3q-22 + 4q-20 + 5q-18 + 3q-16 + 5q-14 + q-12 + 2q-10 + q-8 + 1

HOMFLY-PT Polynomial: a2 + 3a2z2 + a2z4 + a4z-2 + a4 - 2a4z2 - 4a4z4 - a4z6 - 2a6z-2 - 3a6 + a8z-2 + 2a8 + 2a8z2 - a10

Kauffman Polynomial: - a2 + 4a2z2 - 4a2z4 + a2z6 - a3z + 6a3z3 - 8a3z5 + 2a3z7 - a4z-2 + 3a4 - a4z2 - a4z4 - 3a4z6 + a4z8 + 2a5z-1 - 8a5z + 14a5z3 - 11a5z5 + 2a5z7 - 2a6z-2 + 11a6 - 24a6z2 + 27a6z4 - 14a6z6 + 2a6z8 + 2a7z-1 - 12a7z + 16a7z3 - a7z5 - 5a7z7 + a7z9 - a8z-2 + 11a8 - 29a8z2 + 40a8z4 - 21a8z6 + 3a8z8 - 7a9z + 14a9z3 - 3a9z5 - 4a9z7 + a9z9 + 3a10 - 10a10z2 + 16a10z4 - 11a10z6 + 2a10z8 - 2a11z + 6a11z3 - 5a11z5 + a11z7

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 1           1
j = -1          1 
j = -3         21 
j = -5       122  
j = -7      121   
j = -9     222    
j = -11    252     
j = -13   113      
j = -15  121       
j = -17 11         
j = -19 1          
j = -211           


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]=   
3
In[3]:=
Show[DrawMorseLink[Link[11, NonAlternating, 348]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 348]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[11, 20, 12, 21], X[7, 18, 8, 19], 
 
>   X[17, 22, 18, 13], X[9, 17, 10, 16], X[15, 11, 16, 10], X[19, 12, 20, 5], 
 
>   X[21, 8, 22, 9], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, -4, 9, -6, 7, -3, 8}, 
 
>   {11, -2, -7, 6, -5, 4, -8, 3, -9, 5}]
In[6]:=
Jones[L][q]
Out[6]=   
     -10   2    2    2     -6   2    2    3    2
1 - q    + -- - -- + -- + q   + -- - -- + -- - -
            9    8    7          4    3    2   q
           q    q    q          q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -32    -30    -28    3     3     4     5     3     5     -12    2     -8
1 - q    - q    + q    + --- + --- + --- + --- + --- + --- + q    + --- + q
                          24    22    20    18    16    14           10
                         q     q     q     q     q     q            q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 348]][a, z]
Out[8]=   
                               4      6    8
 2    4      6      8    10   a    2 a    a       2  2      4  2      8  2
a  + a  - 3 a  + 2 a  - a   + -- - ---- + -- + 3 a  z  - 2 a  z  + 2 a  z  + 
                               2     2     2
                              z     z     z
 
     2  4      4  4    4  6
>   a  z  - 4 a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 348]][a, z]
Out[9]=   
                                      4      6    8      5      7
  2      4       6       8      10   a    2 a    a    2 a    2 a     3
-a  + 3 a  + 11 a  + 11 a  + 3 a   - -- - ---- - -- + ---- + ---- - a  z - 
                                      2     2     2    z      z
                                     z     z     z
 
       5         7        9        11        2  2    4  2       6  2
>   8 a  z - 12 a  z - 7 a  z - 2 a   z + 4 a  z  - a  z  - 24 a  z  - 
 
        8  2       10  2      3  3       5  3       7  3       9  3
>   29 a  z  - 10 a   z  + 6 a  z  + 14 a  z  + 16 a  z  + 14 a  z  + 
 
       11  3      2  4    4  4       6  4       8  4       10  4      3  5
>   6 a   z  - 4 a  z  - a  z  + 27 a  z  + 40 a  z  + 16 a   z  - 8 a  z  - 
 
        5  5    7  5      9  5      11  5    2  6      4  6       6  6
>   11 a  z  - a  z  - 3 a  z  - 5 a   z  + a  z  - 3 a  z  - 14 a  z  - 
 
        8  6       10  6      3  7      5  7      7  7      9  7    11  7
>   21 a  z  - 11 a   z  + 2 a  z  + 2 a  z  - 5 a  z  - 4 a  z  + a   z  + 
 
     4  8      6  8      8  8      10  8    7  9    9  9
>   a  z  + 2 a  z  + 3 a  z  + 2 a   z  + a  z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    2      1        1        1        1        1        2        1
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 5    3    21  9    19  8    17  8    17  7    15  7    15  6    13  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      1        1        2        3        5        2       2        2
>   ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- + 
     15  5    13  5    11  5    13  4    11  4    9  4    11  3    9  3
    q   t    q   t    q   t    q   t    q   t    q  t    q   t    q  t
 
      1       2       2       1      1      2     t    t      2
>   ----- + ----- + ----- + ----- + ---- + ---- + -- + - + q t
     7  3    9  2    7  2    5  2    7      5      3   q
    q  t    q  t    q  t    q  t    q  t   q  t   q


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