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L11a219

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Acknowledgement

L11a219 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-5/2 + 3q-3/2 - 6q-1/2 + 10q1/2 - 14q3/2 + 15q5/2 - 17q7/2 + 14q9/2 - 11q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant: q-8 - q-4 + q-2 - 3 + q2 + 2q4 + 6q8 + 4q12 + q14 - q16 + 2q18 - 3q20 - q26

HOMFLY-PT Polynomial: a-7z-1 + 2a-7z + a-7z3 - 3a-5z-1 - 6a-5z - 6a-5z3 - 2a-5z5 + 2a-3z-1 + 7a-3z + 7a-3z3 + 4a-3z5 + a-3z7 - 4a-1z - 6a-1z3 - 2a-1z5 + 2az + az3

Kauffman Polynomial: a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + a-8 - 5a-8z2 + 7a-8z4 - 6a-8z6 - a-7z-1 + 5a-7z - 11a-7z3 + 12a-7z5 - 8a-7z7 + 3a-6 - 7a-6z2 + 3a-6z4 + 7a-6z6 - 7a-6z8 - 3a-5z-1 + 10a-5z - 21a-5z3 + 22a-5z5 - 3a-5z7 - 4a-5z9 + 3a-4 + 4a-4z2 - 24a-4z4 + 33a-4z6 - 11a-4z8 - a-4z10 - 2a-3z-1 + 5a-3z - 12a-3z3 + 5a-3z5 + 13a-3z7 - 7a-3z9 + 11a-2z2 - 34a-2z4 + 32a-2z6 - 7a-2z8 - a-2z10 + 2a-1z - 9a-1z3 + 2a-1z5 + 7a-1z7 - 3a-1z9 + 6z2 - 15z4 + 12z6 - 3z8 + 2az - 5az3 + 4az5 - az7

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 18           1
j = 16          2 
j = 14         51 
j = 12        62  
j = 10       85   
j = 8      96    
j = 6     79     
j = 4    78      
j = 2   48       
j = 0  26        
j = -2 14         
j = -4 2          
j = -61           


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, 219]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 219]]
Out[4]=   
PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[20, 10, 21, 9], X[22, 12, 7, 11], 
 
>   X[10, 22, 11, 21], X[2, 7, 3, 8], X[18, 15, 19, 16], X[6, 14, 1, 13], 
 
>   X[4, 18, 5, 17], X[16, 6, 17, 5], X[14, 19, 15, 20]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -6, 2, -9, 10, -8}, 
 
>   {6, -1, 3, -5, 4, -2, 8, -11, 7, -10, 9, -7, 11, -3, 5, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(5/2)    3        6                       3/2       5/2       7/2
-q       + ---- - ------- + 10 Sqrt[q] - 14 q    + 15 q    - 17 q    + 
            3/2   Sqrt[q]
           q
 
        9/2       11/2      13/2      15/2    17/2
>   14 q    - 11 q     + 7 q     - 3 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -8    -4    -2    2      4      8      12    14    16      18      20
-3 + q   - q   + q   + q  + 2 q  + 6 q  + 4 q   + q   - q   + 2 q   - 3 q   - 
 
     26
>   q
In[8]:=
HOMFLYPT[Link[11, Alternating, 219]][a, z]
Out[8]=   
                                                      3      3      3      3
 1      3      2     2 z   6 z   7 z   4 z           z    6 z    7 z    6 z
---- - ---- + ---- + --- - --- + --- - --- + 2 a z + -- - ---- + ---- - ---- + 
 7      5      3      7     5     3     a             7     5      3     a
a  z   a  z   a  z   a     a     a                   a     a      a
 
              5      5      5    7
       3   2 z    4 z    2 z    z
>   a z  - ---- + ---- - ---- + --
             5      3     a      3
            a      a            a
In[9]:=
Kauffman[Link[11, Alternating, 219]][a, z]
Out[9]=   
 -8   3    3     1      3      2     5 z   10 z   5 z   2 z              2
a   + -- + -- - ---- - ---- - ---- + --- + ---- + --- + --- + 2 a z + 6 z  + 
       6    4    7      5      3      7      5     3     a
      a    a    a  z   a  z   a  z   a      a     a
 
     2       2      2      2       2      3       3       3       3      3
    z     5 z    7 z    4 z    11 z    2 z    11 z    21 z    12 z    9 z
>   --- - ---- - ---- + ---- + ----- + ---- - ----- - ----- - ----- - ---- - 
     10     8      6      4      2       9      7       5       3      a
    a      a      a      a      a       a      a       a       a
 
                      4       4      4       4       4      5       5       5
         3       4   z     7 z    3 z    24 z    34 z    3 z    12 z    22 z
>   5 a z  - 15 z  - --- + ---- + ---- - ----- - ----- - ---- + ----- + ----- + 
                      10     8      6      4       2       9      7       5
                     a      a      a      a       a       a      a       a
 
       5      5                       6      6       6       6      7      7
    5 z    2 z         5       6   6 z    7 z    33 z    32 z    8 z    3 z
>   ---- + ---- + 4 a z  + 12 z  - ---- + ---- + ----- + ----- - ---- - ---- + 
      3     a                        8      6      4       2       7      5
     a                              a      a      a       a       a      a
 
        7      7                    8       8      8      9      9      9
    13 z    7 z       7      8   7 z    11 z    7 z    4 z    7 z    3 z
>   ----- + ---- - 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]=   
                                                          2
   2      4     1       2       1     2      4     6   4 q       4        6
8 q  + 7 q  + ----- + ----- + ----- + -- + ----- + - + ---- + 8 q  t + 7 q  t + 
               6  4    4  3    2  3    2    2  2   t    t
              q  t    q  t    q  t    t    q  t
 
       6  2      8  2      8  3      10  3      10  4      12  4      12  5
>   9 q  t  + 9 q  t  + 6 q  t  + 8 q   t  + 5 q   t  + 6 q   t  + 2 q   t  + 
 
       14  5    14  6      16  6    18  7
>   5 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a219
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