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L11a217

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Acknowledgement

L11a217 as Morse Link
DrawMorseLink

PD Presentation: X8192 X12,3,13,4 X16,6,17,5 X22,13,7,14 X18,15,19,16 X14,21,15,22 X20,10,21,9 X4,18,5,17 X10,20,11,19 X2738 X6,11,1,12

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

Jones Polynomial: q-15/2 - 4q-13/2 + 9q-11/2 - 15q-9/2 + 19q-7/2 - 23q-5/2 + 22q-3/2 - 19q-1/2 + 14q1/2 - 8q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant: - q-24 + 3q-20 - 3q-18 + 2q-16 + 4q-14 - 4q-12 + 4q-10 - q-8 + q-6 + 2q-4 - 3q-2 + 5 - 3q2 + 4q6 - 2q8 + q12

HOMFLY-PT Polynomial: - a-3z - a-1z-1 - a-1z + 2a-1z3 + 2az-1 + 5az + 3az3 - az5 - 2a3z-1 - 5a3z - 3a3z3 - 2a3z5 + a5z-1 + 2a5z + 3a5z3 - a7z

Kauffman Polynomial: - a-3z + 2a-3z3 - a-3z5 - a-2z2 + 4a-2z4 - 3a-2z6 - a-1z-1 + 4a-1z - 7a-1z3 + 9a-1z5 - 6a-1z7 + 7z2 - 13z4 + 13z6 - 8z8 - 2az-1 + 12az - 23az3 + 15az5 + az7 - 6az9 - a2 + 13a2z2 - 37a2z4 + 39a2z6 - 14a2z8 - 2a2z10 - 2a3z-1 + 10a3z - 22a3z3 + 9a3z5 + 16a3z7 - 12a3z9 + 8a4z2 - 30a4z4 + 39a4z6 - 13a4z8 - 2a4z10 - a5z-1 + 5a5z - 14a5z3 + 13a5z5 + 5a5z7 - 6a5z9 + 2a6z2 - 8a6z4 + 15a6z6 - 7a6z8 + 2a7z - 6a7z3 + 9a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8           1
j = 6          2 
j = 4         61 
j = 2        82  
j = 0       116   
j = -2      129    
j = -4     1110     
j = -6    913      
j = -8   610       
j = -10  39        
j = -12 16         
j = -14 3          
j = -161           


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, 217]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 217]]
Out[4]=   
PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[16, 6, 17, 5], X[22, 13, 7, 14], 
 
>   X[18, 15, 19, 16], X[14, 21, 15, 22], X[20, 10, 21, 9], X[4, 18, 5, 17], 
 
>   X[10, 20, 11, 19], X[2, 7, 3, 8], X[6, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -8, 3, -11}, 
 
>   {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 9, -7, 6, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     4       9      15     19     23     22      19
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
 
       3/2      5/2    7/2
>   8 q    + 3 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    3     3     2     4     4     4     -8    -6   2    3       2
5 - q    + --- - --- + --- + --- - --- + --- - q   + q   + -- - -- - 3 q  + 
            20    18    16    14    12    10                4    2
           q     q     q     q     q     q                 q    q
 
       6      8    12
>   4 q  - 2 q  + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 217]][a, z]
Out[8]=   
                  3    5                                                3
   1     2 a   2 a    a    z    z              3        5      7     2 z
-(---) + --- - ---- + -- - -- - - + 5 a z - 5 a  z + 2 a  z - a  z + ---- + 
  a z     z     z     z     3   a                                     a
                           a
 
         3      3  3      5  3      5      3  5
>   3 a z  - 3 a  z  + 3 a  z  - a z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 217]][a, z]
Out[9]=   
                     3    5
  2    1    2 a   2 a    a    z    4 z                3        5        7
-a  - --- - --- - ---- - -- - -- + --- + 12 a z + 10 a  z + 5 a  z + 2 a  z + 
      a z    z     z     z     3    a
                              a
 
            2                                             3      3
       2   z        2  2      4  2      6  2    8  2   2 z    7 z          3
>   7 z  - -- + 13 a  z  + 8 a  z  + 2 a  z  - a  z  + ---- - ---- - 23 a z  - 
            2                                            3     a
           a                                            a
 
                                               4
        3  3       5  3      7  3       4   4 z        2  4       4  4
>   22 a  z  - 14 a  z  - 6 a  z  - 13 z  + ---- - 37 a  z  - 30 a  z  - 
                                              2
                                             a
 
                         5      5
       6  4      8  4   z    9 z          5      3  5       5  5      7  5
>   8 a  z  + 2 a  z  - -- + ---- + 15 a z  + 9 a  z  + 13 a  z  + 9 a  z  + 
                         3    a
                        a
 
               6                                               7
        6   3 z        2  6       4  6       6  6    8  6   6 z       7
>   13 z  - ---- + 39 a  z  + 39 a  z  + 15 a  z  - a  z  - ---- + a z  + 
              2                                              a
             a
 
        3  7      5  7      7  7      8       2  8       4  8      6  8
>   16 a  z  + 5 a  z  - 4 a  z  - 8 z  - 14 a  z  - 13 a  z  - 7 a  z  - 
 
         9       3  9      5  9      2  10      4  10
>   6 a z  - 12 a  z  - 6 a  z  - 2 a  z   - 2 a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     9      1        3        1        6        3        9        6      10
11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
      2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
     q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      9      13      11      10     12             2        2  2      4  2
>   ----- + ----- + ----- + ---- + ---- + 6 t + 8 q  t + 2 q  t  + 6 q  t  + 
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t
 
     4  3      6  3    8  4
>   q  t  + 2 q  t  + q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a217
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