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L11a447

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Acknowledgement

L11a447 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-4 + 3q-3 - 6q-2 + 10q-1 - 11 + 15q - 14q2 + 13q3 - 9q4 + 6q5 - 3q6 + q7

A2 (sl(3)) Invariant: - q-12 - q-6 + 3q-4 + q-2 + 5 + 5q2 + 2q4 + 6q6 + 4q10 + q12 + 2q16 - q18 + q20

HOMFLY-PT Polynomial: a-4z-2 + 3a-4 + 5a-4z2 + 4a-4z4 + a-4z6 - 2a-2z-2 - 9a-2 - 16a-2z2 - 14a-2z4 - 6a-2z6 - a-2z8 + z-2 + 8 + 13z2 + 9z4 + 2z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial: - a-8z2 + a-8z4 - 3a-7z3 + 3a-7z5 - a-6 + 3a-6z2 - 6a-6z4 + 5a-6z6 - a-5z + 5a-5z3 - 8a-5z5 + 6a-5z7 - a-4z-2 + 3a-4 - 5a-4z2 + 8a-4z4 - 10a-4z6 + 6a-4z8 + 2a-3z-1 - 8a-3z + 16a-3z3 - 10a-3z5 - 3a-3z7 + 4a-3z9 - 2a-2z-2 + 11a-2 - 32a-2z2 + 52a-2z4 - 41a-2z6 + 10a-2z8 + a-2z10 + 2a-1z-1 - 12a-1z + 18a-1z3 - 17a-1z7 + 7a-1z9 - z-2 + 11 - 32z2 + 52z4 - 38z6 + 7z8 + z10 - 7az + 15az3 - 5az5 - 7az7 + 3az9 + 3a2 - 9a2z2 + 15a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 15           1
j = 13          2 
j = 11         41 
j = 9        63  
j = 7       73   
j = 5      76    
j = 3     87     
j = 1    610      
j = -1   45       
j = -3  26        
j = -5 14         
j = -7 2          
j = -91           


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, 447]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 447]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[8, 18, 9, 17], X[16, 8, 17, 7], 
 
>   X[18, 10, 19, 9], X[10, 14, 11, 13], X[20, 11, 21, 12], X[22, 19, 13, 20], 
 
>   X[12, 21, 5, 22], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6, 7, -9}, 
 
>   {6, -2, 11, -4, 3, -5, 8, -7, 9, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
       -4   3    6    10              2       3      4      5      6    7
-11 - q   + -- - -- + -- + 15 q - 14 q  + 13 q  - 9 q  + 6 q  - 3 q  + q
             3    2   q
            q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -12    -6   3     -2      2      4      6      10    12      16    18    20
5 - q    - q   + -- + q   + 5 q  + 2 q  + 6 q  + 4 q   + q   + 2 q   - q   + q
                  4
                 q
In[8]:=
HOMFLYPT[Link[11, Alternating, 447]][a, z]
Out[8]=   
                                                      2       2
    3    9       2    -2     1       2         2   5 z    16 z       2  2
8 + -- - -- - 2 a  + z   + ----- - ----- + 13 z  + ---- - ----- - 3 a  z  + 
     4    2                 4  2    2  2             4      2
    a    a                 a  z    a  z             a      a
 
              4       4                   6      6    8
       4   4 z    14 z     2  4      6   z    6 z    z
>   9 z  + ---- - ----- - a  z  + 2 z  + -- - ---- - --
             4      2                     4     2     2
            a      a                     a     a     a
In[9]:=
Kauffman[Link[11, Alternating, 447]][a, z]
Out[9]=   
      -6   3    11      2    -2     1       2      2      2    z    8 z
11 - a   + -- + -- + 3 a  - z   - ----- - ----- + ---- + --- - -- - --- - 
            4    2                 4  2    2  2    3     a z    5    3
           a    a                 a  z    a  z    a  z         a    a
 
                                     2      2      2       2                3
    12 z              3         2   z    3 z    5 z    32 z       2  2   3 z
>   ---- - 7 a z - 2 a  z - 32 z  - -- + ---- - ---- - ----- - 9 a  z  - ---- + 
     a                               8     6      4      2                 7
                                    a     a      a      a                 a
 
       3       3       3                                4      4      4
    5 z    16 z    18 z          3      3  3       4   z    6 z    8 z
>   ---- + ----- + ----- + 15 a z  + 5 a  z  + 52 z  + -- - ---- + ---- + 
      5      3       a                                  8     6      4
     a      a                                          a     a      a
 
        4                 5      5       5                                 6
    52 z        2  4   3 z    8 z    10 z         5      3  5       6   5 z
>   ----- + 15 a  z  + ---- - ---- - ----- - 5 a z  - 4 a  z  - 38 z  + ---- - 
      2                  7      5      3                                  6
     a                  a      a      a                                  a
 
        6       6                 7      7       7
    10 z    41 z        2  6   6 z    3 z    17 z         7    3  7      8
>   ----- - ----- - 12 a  z  + ---- - ---- - ----- - 7 a z  + a  z  + 7 z  + 
      4       2                  5      3      a
     a       a                  a      a
 
       8       8                9      9                   10
    6 z    10 z       2  8   4 z    7 z         9    10   z
>   ---- + ----- + 3 a  z  + ---- + ---- + 3 a z  + z   + ---
      4      2                 3     a                     2
     a      a                 a                           a
In[10]:=
Kh[L][q, t]
Out[10]=   
          3     1       2       1       4       2       6      4      5
10 q + 8 q  + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 
               9  5    7  4    5  4    5  3    3  3    3  2      2   q t
              q  t    q  t    q  t    q  t    q  t    q  t    q t
 
    6 q      3        5        5  2      7  2      7  3      9  3      9  4
>   --- + 7 q  t + 7 q  t + 6 q  t  + 7 q  t  + 3 q  t  + 6 q  t  + 3 q  t  + 
     t
 
       11  4    11  5      13  5    15  6
>   4 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a447
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