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L11a436

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Acknowledgement

L11a436 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X18,11,19,12 X14,8,15,7 X8,14,9,13 X20,15,21,16 X22,17,13,18 X16,21,17,22 X12,19,5,20 X2536 X4,9,1,10

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

Jones Polynomial: - q-11 + 3q-10 - 8q-9 + 13q-8 - 16q-7 + 19q-6 - 17q-5 + 16q-4 - 10q-3 + 6q-2 - 2q-1 + 1

A2 (sl(3)) Invariant: - q-34 - q-32 - 5q-28 + 8q-20 + 3q-18 + 9q-16 + 4q-14 + 2q-12 + 5q-10 - 2q-8 + 3q-6 + q-4 + 1

HOMFLY-PT Polynomial: 2a2 + 3a2z2 + a2z4 + 2a4z-2 + 2a4 - 2a4z4 - a4z6 - 5a6z-2 - 10a6 - 10a6z2 - 7a6z4 - 2a6z6 + 4a8z-2 + 8a8 + 8a8z2 + 3a8z4 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial: - 2a2 + 5a2z2 - 4a2z4 + a2z6 + a3z + 2a3z3 - 5a3z5 + 2a3z7 - 2a4z-2 + 6a4 - 6a4z2 + 4a4z4 - 6a4z6 + 3a4z8 + 5a5z-1 - 15a5z + 14a5z3 - 5a5z5 - 3a5z7 + 3a5z9 - 5a6z-2 + 22a6 - 44a6z2 + 49a6z4 - 32a6z6 + 9a6z8 + a6z10 + 9a7z-1 - 35a7z + 43a7z3 - 18a7z5 - 6a7z7 + 7a7z9 - 4a8z-2 + 21a8 - 45a8z2 + 55a8z4 - 39a8z6 + 13a8z8 + a8z10 + 5a9z-1 - 25a9z + 41a9z3 - 29a9z5 + 5a9z7 + 4a9z9 - a10z-2 + 6a10 - 11a10z2 + 10a10z4 - 11a10z6 + 7a10z8 + a11z-1 - 5a11z + 8a11z3 - 10a11z5 + 6a11z7 + a12z2 - 4a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 1           1
j = -1          1 
j = -3         51 
j = -5        73  
j = -7       93   
j = -9      87    
j = -11     119     
j = -13    710      
j = -15   69       
j = -17  27        
j = -19 16         
j = -21 2          
j = -231           


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, Alternating, 436]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 436]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[14, 8, 15, 7], 
 
>   X[8, 14, 9, 13], X[20, 15, 21, 16], X[22, 17, 13, 18], X[16, 21, 17, 22], 
 
>   X[12, 19, 5, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, 
 
>   {5, -4, 6, -8, 7, -3, 9, -6, 8, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
     -11    3    8    13   16   19   17   16   10   6    2
1 - q    + --- - -- + -- - -- + -- - -- + -- - -- + -- - -
            10    9    8    7    6    5    4    3    2   q
           q     q    q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -34    -32    5     8     3     9     4     2     5    2    3     -4
1 - q    - q    - --- + --- + --- + --- + --- + --- + --- - -- + -- + q
                   28    20    18    16    14    12    10    8    6
                  q     q     q     q     q     q     q     q    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 436]][a, z]
Out[8]=   
                                        4      6      8    10
   2      4       6      8      10   2 a    5 a    4 a    a        2  2
2 a  + 2 a  - 10 a  + 8 a  - 2 a   + ---- - ---- + ---- - --- + 3 a  z  - 
                                       2      2      2     2
                                      z      z      z     z
 
        6  2      8  2    10  2    2  4      4  4      6  4      8  4    4  6
>   10 a  z  + 8 a  z  - a   z  + a  z  - 2 a  z  - 7 a  z  + 3 a  z  - a  z  - 
 
       6  6
>   2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 436]][a, z]
Out[9]=   
                                          4      6      8    10      5      7
    2      4       6       8      10   2 a    5 a    4 a    a     5 a    9 a
-2 a  + 6 a  + 22 a  + 21 a  + 6 a   - ---- - ---- - ---- - --- + ---- + ---- + 
                                         2      2      2     2     z      z
                                        z      z      z     z
 
       9    11
    5 a    a      3         5         7         9        11      13
>   ---- + --- + a  z - 15 a  z - 35 a  z - 25 a  z - 5 a   z + a   z + 
     z      z
 
       2  2      4  2       6  2       8  2       10  2    12  2      3  3
>   5 a  z  - 6 a  z  - 44 a  z  - 45 a  z  - 11 a   z  + a   z  + 2 a  z  + 
 
        5  3       7  3       9  3      11  3      13  3      2  4      4  4
>   14 a  z  + 43 a  z  + 41 a  z  + 8 a   z  - 2 a   z  - 4 a  z  + 4 a  z  + 
 
        6  4       8  4       10  4      12  4      3  5      5  5       7  5
>   49 a  z  + 55 a  z  + 10 a   z  - 4 a   z  - 5 a  z  - 5 a  z  - 18 a  z  - 
 
        9  5       11  5    13  5    2  6      4  6       6  6       8  6
>   29 a  z  - 10 a   z  + a   z  + a  z  - 6 a  z  - 32 a  z  - 39 a  z  - 
 
        10  6      12  6      3  7      5  7      7  7      9  7      11  7
>   11 a   z  + 3 a   z  + 2 a  z  - 3 a  z  - 6 a  z  + 5 a  z  + 6 a   z  + 
 
       4  8      6  8       8  8      10  8      5  9      7  9      9  9
>   3 a  z  + 9 a  z  + 13 a  z  + 7 a   z  + 3 a  z  + 7 a  z  + 4 a  z  + 
 
     6  10    8  10
>   a  z   + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3    5      1        2        1        6        2        7        6
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 5    3    23  9    21  8    19  8    19  7    17  7    17  6    15  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      9        7        10       11       9        8       7       9      3
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 
     15  5    13  5    13  4    11  4    11  3    9  3    9  2    7  2    7
    q   t    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
     7     t    t      2
>   ---- + -- + - + q t
     5      3   q
    q  t   q


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