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L11a68

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Acknowledgement

L11a68 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-1/2 + 2q1/2 - 6q3/2 + 10q5/2 - 15q7/2 + 16q9/2 - 17q11/2 + 15q13/2 - 11q15/2 + 7q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant: q-2 + q2 + 4q4 - 2q6 + 3q8 + 2q10 - q12 + 4q14 - q16 + 3q18 - 2q22 + 2q24 - 4q26 - q28 + q30 - q32

HOMFLY-PT Polynomial: a-9z-1 + a-9z + a-9z3 - 2a-7z-1 - 2a-7z - a-7z3 - a-7z5 + a-5z-1 - 3a-5z3 - 2a-5z5 - a-3z-1 - a-3z - a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3

Kauffman Polynomial: - 2a-12z2 + 3a-12z4 - a-12z6 - 5a-11z3 + 8a-11z5 - 3a-11z7 - 3a-10 + 9a-10z2 - 14a-10z4 + 14a-10z6 - 5a-10z8 + a-9z-1 - 3a-9z + 6a-9z3 - 4a-9z5 + 7a-9z7 - 4a-9z9 - 7a-8 + 26a-8z2 - 37a-8z4 + 30a-8z6 - 9a-8z8 - a-8z10 + 2a-7z-1 - 6a-7z + 10a-7z3 - 12a-7z5 + 13a-7z7 - 7a-7z9 - 4a-6 + 17a-6z2 - 28a-6z4 + 22a-6z6 - 8a-6z8 - a-6z10 + a-5z-1 - 3a-5z - a-5z3 + 3a-5z5 - 3a-5z9 + 2a-4z2 - 5a-4z4 + 5a-4z6 - 4a-4z8 + a-3z-1 - 3a-3z + 3a-3z3 + 2a-3z5 - 3a-3z7 - a-2 + 3a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5

Khovanov Homology:
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 22           1
j = 20          2 
j = 18         51 
j = 16        62  
j = 14       95   
j = 12      86    
j = 10     89     
j = 8    78      
j = 6   38       
j = 4  37        
j = 2 15         
j = 0 1          
j = -21           


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, 68]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 68]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[20, 12, 21, 11], 
 
>   X[22, 18, 5, 17], X[18, 22, 19, 21], X[10, 20, 11, 19], X[14, 10, 15, 9], 
 
>   X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 8, -7, 4, -2, 11, -8, 9, -3, 5, -6, 
 
>    7, -4, 6, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
     1                      3/2       5/2       7/2       9/2       11/2
-(-------) + 2 Sqrt[q] - 6 q    + 10 q    - 15 q    + 16 q    - 17 q     + 
  Sqrt[q]
 
        13/2       15/2      17/2      19/2    21/2
>   15 q     - 11 q     + 7 q     - 3 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -2    2      4      6      8      10    12      14    16      18      22
q   + q  + 4 q  - 2 q  + 3 q  + 2 q   - q   + 4 q   - q   + 3 q   - 2 q   + 
 
       24      26    28    30    32
>   2 q   - 4 q   - q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 68]][a, z]
Out[8]=   
                                                         3    3      3    3
 1      2      1      1      1    z    2 z   z    2 z   z    z    3 z    z
---- - ---- + ---- - ---- + --- + -- - --- - -- + --- + -- - -- - ---- - -- + 
 9      7      5      3     a z    9    7     3    a     9    7     5     3
a  z   a  z   a  z   a  z         a    a     a          a    a     a     a
 
     3    5      5    5
    z    z    2 z    z
>   -- - -- - ---- - --
    a     7     5     3
         a     a     a
In[9]:=
Kauffman[Link[11, Alternating, 68]][a, z]
Out[9]=   
-3    7    4     -2    1      2      1      1      1    3 z   6 z   3 z   3 z
--- - -- - -- - a   + ---- + ---- + ---- + ---- + --- - --- - --- - --- - --- - 
 10    8    6          9      7      5      3     a z    9     7     5     3
a     a    a          a  z   a  z   a  z   a  z         a     a     a     a
 
             2      2       2       2      2      3      3       3    3
    3 z   2 z    9 z    26 z    17 z    2 z    5 z    6 z    10 z    z
>   --- - ---- + ---- + ----- + ----- + ---- - ---- + ---- + ----- - -- + 
     a     12     10      8       6       4     11      9      7      5
          a      a       a       a       a     a       a      a      a
 
       3      3      4       4       4       4      4      4      5      5
    3 z    3 z    3 z    14 z    37 z    28 z    5 z    3 z    8 z    4 z
>   ---- + ---- + ---- - ----- - ----- - ----- - ---- + ---- + ---- - ---- - 
      3     a      12      10      8       6       4      2     11      9
     a            a       a       a       a       a      a     a       a
 
        5      5      5    5    6        6       6       6      6      6
    12 z    3 z    2 z    z    z     14 z    30 z    22 z    5 z    2 z
>   ----- + ---- + ---- - -- - --- + ----- + ----- + ----- + ---- - ---- - 
      7       5      3    a     12     10      8       6       4      2
     a       a      a          a      a       a       a       a      a
 
       7      7       7      7      8      8      8      8      9      9
    3 z    7 z    13 z    3 z    5 z    9 z    8 z    4 z    4 z    7 z
>   ---- + ---- + ----- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - 
     11      9      7       3     10      8      6      4      9      7
    a       a      a       a     a       a      a      a      a      a
 
       9    10    10
    3 z    z     z
>   ---- - --- - ---
      5     8     6
     a     a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
                           2
   2      4     1     1   q       4        6        6  2      8  2      8  3
5 q  + 3 q  + ----- + - + -- + 7 q  t + 3 q  t + 8 q  t  + 7 q  t  + 8 q  t  + 
               2  2   t   t
              q  t
 
       10  3      10  4      12  4      12  5      14  5      14  6
>   8 q   t  + 9 q   t  + 8 q   t  + 6 q   t  + 9 q   t  + 5 q   t  + 
 
       16  6      16  7      18  7    18  8      20  8    22  9
>   6 q   t  + 2 q   t  + 5 q   t  + q   t  + 2 q   t  + q   t


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