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L11a499

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Acknowledgement

L11a499 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-3 - 2q-2 + 7q-1 - 11 + 16q - 17q2 + 19q3 - 16q4 + 12q5 - 7q6 + 3q7 - q8

A2 (sl(3)) Invariant: q-10 + q-8 + q-6 + 6q-4 + 2q-2 + 3 + 6q2 - q4 + 5q6 + 2q10 + 3q12 - 2q14 + 4q16 - 2q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial: - 2a-6 - 2a-6z2 - a-6z4 + 4a-4 + 6a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 - a-2z2 + a-2z4 + a-2z6 - 2z-2 - 4 - 4z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial: - 2a-9z3 + a-9z5 - 5a-8z4 + 3a-8z6 - 4a-7z + 11a-7z3 - 13a-7z5 + 6a-7z7 + 4a-6 - 14a-6z2 + 27a-6z4 - 21a-6z6 + 8a-6z8 - 8a-5z + 21a-5z3 - 9a-5z5 - 4a-5z7 + 5a-5z9 + 8a-4 - 36a-4z2 + 62a-4z4 - 46a-4z6 + 14a-4z8 + a-4z10 - 3a-3z3 + 9a-3z5 - 14a-3z7 + 8a-3z9 + a-2z-2 - 16a-2z2 + 27a-2z4 - 26a-2z6 + 9a-2z8 + a-2z10 - 2a-1z-1 + 8a-1z - 11a-1z3 - 2a-1z7 + 3a-1z9 + 2z-2 - 7 + 12z2 - 7z4 - 3z6 + 3z8 - 2az-1 + 4az - 4az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 17           1
j = 15          2 
j = 13         51 
j = 11        72  
j = 9       95   
j = 7      107    
j = 5     79     
j = 3    910      
j = 1   611       
j = -1  15        
j = -3 16         
j = -5 1          
j = -71           


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, 499]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 499]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 8, 17, 7], X[22, 17, 19, 18], 
 
>   X[20, 12, 21, 11], X[10, 20, 11, 19], X[18, 21, 5, 22], 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}, {6, -5, 7, -4}, 
 
>   {10, -1, 3, -9, 8, -6, 5, -2, 11, -8, 9, -3, 4, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
       -3   2    7              2       3       4       5      6      7    8
-11 + q   - -- + - + 16 q - 17 q  + 19 q  - 16 q  + 12 q  - 7 q  + 3 q  - q
             2   q
            q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -10    -8    -6   6    2       2    4      6      10      12      14
3 + q    + q   + q   + -- + -- + 6 q  - q  + 5 q  + 2 q   + 3 q   - 2 q   + 
                        4    2
                       q    q
 
       16      18      20    22    24
>   4 q   - 2 q   - 2 q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 499]][a, z]
Out[8]=   
                                    2             2      2    2
     2    4       2   2      1     a       2   2 z    6 z    z     2  2
-4 - -- + -- + 2 a  - -- + ----- + -- - 4 z  - ---- + ---- - -- + a  z  - 
      6    4           2    2  2    2            6      4     2
     a    a           z    a  z    z            a      a     a
 
            4      4    4    6    6
       4   z    3 z    z    z    z
>   2 z  - -- + ---- + -- + -- + --
            6     4     2    4    2
           a     a     a    a    a
In[9]:=
Kauffman[Link[11, Alternating, 499]][a, z]
Out[9]=   
                                    2
     4    8       2   2      1     a     2    2 a   4 z   8 z   8 z
-7 + -- + -- - 4 a  + -- + ----- + -- - --- - --- - --- - --- + --- + 4 a z + 
      6    4           2    2  2    2   a z    z     7     5     a
     a    a           z    a  z    z                a     a
 
                2       2       2                3       3       3      3
        2   14 z    36 z    16 z       2  2   2 z    11 z    21 z    3 z
>   12 z  - ----- - ----- - ----- + 6 a  z  - ---- + ----- + ----- - ---- - 
              6       4       2                 9      7       5       3
             a       a       a                 a      a       a       a
 
        3             4       4       4       4              5       5      5
    11 z       4   5 z    27 z    62 z    27 z       2  4   z    13 z    9 z
>   ----- - 7 z  - ---- + ----- + ----- + ----- - 4 a  z  + -- - ----- - ---- + 
      a              8      6       4       2                9     7       5
                    a      a       a       a                a     a       a
 
       5                      6       6       6       6              7      7
    9 z         5      6   3 z    21 z    46 z    26 z     2  6   6 z    4 z
>   ---- - 4 a z  - 3 z  + ---- - ----- - ----- - ----- + a  z  + ---- - ---- - 
      3                      8      6       4       2               7      5
     a                      a      a       a       a               a      a
 
        7      7                      8       8      8      9      9      9
    14 z    2 z         7      8   8 z    14 z    9 z    5 z    8 z    3 z
>   ----- - ---- + 2 a z  + 3 z  + ---- + ----- + ---- + ---- + ---- + ---- + 
      3      a                       6      4       2      5      3     a
     a                              a      a       a      a      a
 
     10    10
    z     z
>   --- + ---
     4     2
    a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
          3     1       1       1       6      1      5    6 q       3
11 q + 9 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q  t + 
               7  4    5  3    3  3    3  2      2   q t    t
              q  t    q  t    q  t    q  t    q t
 
       5        5  2       7  2      7  3      9  3      9  4      11  4
>   7 q  t + 9 q  t  + 10 q  t  + 7 q  t  + 9 q  t  + 5 q  t  + 7 q   t  + 
 
       11  5      13  5    13  6      15  6    17  7
>   2 q   t  + 5 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a499
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