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L11a107

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Acknowledgement

L11a107 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-29/2 - 2q-27/2 + 4q-25/2 - 5q-23/2 + 7q-21/2 - 8q-19/2 + 7q-17/2 - 7q-15/2 + 4q-13/2 - 4q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant: - q-44 - 2q-42 - q-40 - 2q-38 - q-36 - q-34 + 2q-30 + q-28 + 5q-26 + 3q-24 + 3q-22 + 2q-20 + q-16 - q-14 + q-12

HOMFLY-PT Polynomial: - a7z - 6a7z3 - 5a7z5 - a7z7 - 4a9z-1 - 17a9z - 23a9z3 - 12a9z5 - 2a9z7 + 7a11z-1 + 20a11z + 15a11z3 + 3a11z5 - 3a13z-1 - 4a13z - a13z3

Kauffman Polynomial: a7z - 6a7z3 + 5a7z5 - a7z7 + a8z2 - 9a8z4 + 9a8z6 - 2a8z8 + 4a9z-1 - 18a9z + 35a9z3 - 34a9z5 + 17a9z7 - 3a9z9 - 7a10 + 23a10z2 - 27a10z4 + 12a10z6 + a10z8 - a10z10 + 7a11z-1 - 29a11z + 57a11z3 - 55a11z5 + 27a11z7 - 5a11z9 - 7a12 + 22a12z2 - 24a12z4 + 9a12z6 + a12z8 - a12z10 + 3a13z-1 - 10a13z + 12a13z3 - 12a13z5 + 7a13z7 - 2a13z9 - 4a14z4 + 4a14z6 - 2a14z8 - a15z3 + 2a15z5 - 2a15z7 + 2a16z2 + a16z4 - 2a16z6 + 3a17z3 - 2a17z5 - a18 + 2a18z2 - a18z4

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -6           1
j = -8          21
j = -10         2  
j = -12        22  
j = -14       52   
j = -16      33    
j = -18     54     
j = -20    23      
j = -22   35       
j = -24  12        
j = -26 13         
j = -28 1          
j = -301           


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, 107]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 107]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[22, 15, 5, 16], X[16, 7, 17, 8], 
 
>   X[18, 9, 19, 10], X[20, 11, 21, 12], X[8, 17, 9, 18], X[10, 19, 11, 20], 
 
>   X[12, 21, 13, 22], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 5, -8, 6, -9, 11, -2, 3, -4, 7, -5, 
 
>    8, -6, 9, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(29/2)     2       4       5       7       8       7       7       4
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 
            27/2    25/2    23/2    21/2    19/2    17/2    15/2    13/2
           q       q       q       q       q       q       q       q
 
      4      2      -(7/2)
>   ----- + ---- - q
     11/2    9/2
    q       q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -44    2     -40    2     -36    -34    2     -28    5     3     3     2
-q    - --- - q    - --- - q    - q    + --- + q    + --- + --- + --- + --- + 
         42           38                  30           26    24    22    20
        q            q                   q            q     q     q     q
 
     -16    -14    -12
>   q    - q    + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 107]][a, z]
Out[8]=   
    9      11      13
-4 a    7 a     3 a      7         9         11        13        7  3
----- + ----- - ----- - a  z - 17 a  z + 20 a   z - 4 a   z - 6 a  z  - 
  z       z       z
 
        9  3       11  3    13  3      7  5       9  5      11  5    7  7
>   23 a  z  + 15 a   z  - a   z  - 5 a  z  - 12 a  z  + 3 a   z  - a  z  - 
 
       9  7
>   2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 107]][a, z]
Out[9]=   
                          9      11      13
    10      12    18   4 a    7 a     3 a      7         9         11
-7 a   - 7 a   - a   + ---- + ----- + ----- + a  z - 18 a  z - 29 a   z - 
                        z       z       z
 
        13      8  2       10  2       12  2      16  2      18  2      7  3
>   10 a   z + a  z  + 23 a   z  + 22 a   z  + 2 a   z  + 2 a   z  - 6 a  z  + 
 
        9  3       11  3       13  3    15  3      17  3      8  4
>   35 a  z  + 57 a   z  + 12 a   z  - a   z  + 3 a   z  - 9 a  z  - 
 
        10  4       12  4      14  4    16  4    18  4      7  5       9  5
>   27 a   z  - 24 a   z  - 4 a   z  + a   z  - a   z  + 5 a  z  - 34 a  z  - 
 
        11  5       13  5      15  5      17  5      8  6       10  6
>   55 a   z  - 12 a   z  + 2 a   z  - 2 a   z  + 9 a  z  + 12 a   z  + 
 
       12  6      14  6      16  6    7  7       9  7       11  7      13  7
>   9 a   z  + 4 a   z  - 2 a   z  - a  z  + 17 a  z  + 27 a   z  + 7 a   z  - 
 
       15  7      8  8    10  8    12  8      14  8      9  9      11  9
>   2 a   z  - 2 a  z  + a   z  + a   z  - 2 a   z  - 3 a  z  - 5 a   z  - 
 
       13  9    10  10    12  10
>   2 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -8    -6      1         1         1        3        1        2        3
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             30  11    28  10    26  10    26  9    24  9    24  8    22  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      5        2        3        5        4        3        3        5
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     22  7    20  7    20  6    18  6    18  5    16  5    16  4    14  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      2        2        2        2       2
>   ------ + ------ + ------ + ------ + ----
     14  3    12  3    12  2    10  2    8
    q   t    q   t    q   t    q   t    q  t


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