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L11a109

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Acknowledgement

L11a109 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-27/2 - 3q-25/2 + 7q-23/2 - 11q-21/2 + 15q-19/2 - 17q-17/2 + 17q-15/2 - 16q-13/2 + 10q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant: - q-42 - q-40 + q-38 - 3q-36 - q-34 + q-32 - 3q-30 + 3q-28 + 2q-24 + 4q-22 + 6q-18 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial: - 2a5z3 - a5z5 - 2a7z-1 - 8a7z - 9a7z3 - 3a7z5 + 2a9z-1 + 2a9z - 3a9z3 - 2a9z5 + a11z-1 + 5a11z + 3a11z3 - a13z-1 - a13z

Kauffman Polynomial: 2a5z3 - a5z5 + 5a6z4 - 3a6z6 - 2a7z-1 + 8a7z - 16a7z3 + 15a7z5 - 6a7z7 + 3a8 - 3a8z2 - 7a8z4 + 11a8z6 - 6a8z8 - 2a9z-1 + 6a9z - 14a9z3 + 7a9z5 + 2a9z7 - 4a9z9 + 10a10z2 - 30a10z4 + 28a10z6 - 10a10z8 - a10z10 + a11z-1 - 3a11z + 4a11z3 - 6a11z5 + 11a11z7 - 7a11z9 - 3a12 + 13a12z2 - 20a12z4 + 22a12z6 - 8a12z8 - a12z10 + a13z-1 - 6a13z3 + 11a13z5 - 3a13z9 - 3a14z2 + a14z4 + 7a14z6 - 4a14z8 + a15z - 6a15z3 + 8a15z5 - 3a15z7 + a16 - 3a16z2 + 3a16z4 - a16z6

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4           1
j = -6          31
j = -8         4  
j = -10        63  
j = -12       104   
j = -14      87    
j = -16     99     
j = -18    68      
j = -20   59       
j = -22  26        
j = -24 15         
j = -26 2          
j = -281           


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[11, Alternating, 109]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 109]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[22, 15, 5, 16], X[16, 7, 17, 8], 
 
>   X[8, 21, 9, 22], X[20, 11, 21, 12], X[18, 9, 19, 10], X[10, 19, 11, 20], 
 
>   X[12, 17, 13, 18], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 7, -8, 6, -9, 11, -2, 3, -4, 9, -7, 
 
>    8, -6, 5, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(27/2)     3       7      11      15      17      17      16      10
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 
            25/2    23/2    21/2    19/2    17/2    15/2    13/2    11/2
           q       q       q       q       q       q       q       q
 
     7      3      -(5/2)
>   ---- + ---- - q
     9/2    7/2
    q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    -40    -38    3     -34    -32    3     3     2     4     6     2
-q    - q    + q    - --- - q    + q    - --- + --- + --- + --- + --- + --- - 
                       36                  30    28    24    22    18    12
                      q                   q     q     q     q     q     q
 
     2     -8
>   --- + q
     10
    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 109]][a, z]
Out[8]=   
    7      9    11    13
-2 a    2 a    a     a        7        9        11      13        5  3
----- + ---- + --- - --- - 8 a  z + 2 a  z + 5 a   z - a   z - 2 a  z  - 
  z      z      z     z
 
       7  3      9  3      11  3    5  5      7  5      9  5
>   9 a  z  - 3 a  z  + 3 a   z  - a  z  - 3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 109]][a, z]
Out[9]=   
                        7      9    11    13
   8      12    16   2 a    2 a    a     a        7        9        11
3 a  - 3 a   + a   - ---- - ---- + --- + --- + 8 a  z + 6 a  z - 3 a   z + 
                      z      z      z     z
 
     15        8  2       10  2       12  2      14  2      16  2      5  3
>   a   z - 3 a  z  + 10 a   z  + 13 a   z  - 3 a   z  - 3 a   z  + 2 a  z  - 
 
        7  3       9  3      11  3      13  3      15  3      6  4      8  4
>   16 a  z  - 14 a  z  + 4 a   z  - 6 a   z  - 6 a   z  + 5 a  z  - 7 a  z  - 
 
        10  4       12  4    14  4      16  4    5  5       7  5      9  5
>   30 a   z  - 20 a   z  + a   z  + 3 a   z  - a  z  + 15 a  z  + 7 a  z  - 
 
       11  5       13  5      15  5      6  6       8  6       10  6
>   6 a   z  + 11 a   z  + 8 a   z  - 3 a  z  + 11 a  z  + 28 a   z  + 
 
        12  6      14  6    16  6      7  7      9  7       11  7      15  7
>   22 a   z  + 7 a   z  - a   z  - 6 a  z  + 2 a  z  + 11 a   z  - 3 a   z  - 
 
       8  8       10  8      12  8      14  8      9  9      11  9      13  9
>   6 a  z  - 10 a   z  - 8 a   z  - 4 a   z  - 4 a  z  - 7 a   z  - 3 a   z  - 
 
     10  10    12  10
>   a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         2         1        5        2        6        5
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             28  11    26  10    24  10    24  9    22  9    22  8    20  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      9        6        8        9        9        8        7        10
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     20  7    18  7    18  6    16  6    16  5    14  5    14  4    12  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      4        6        3        4      3
>   ------ + ------ + ------ + ----- + ----
     12  3    10  3    10  2    8  2    6
    q   t    q   t    q   t    q  t    q  t


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