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L11a486

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Acknowledgement

L11a486 as Morse Link
DrawMorseLink

PD Presentation: X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X18,12,5,11 X2,16,3,15

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

Jones Polynomial: - q-2 + 4q-1 - 9 + 17q - 22q2 + 27q3 - 25q4 + 23q5 - 17q6 + 10q7 - 4q8 + q9

A2 (sl(3)) Invariant: - q-6 + 2q-4 - q-2 - 2 + 6q2 - 3q4 + 5q6 + 5q8 + 2q10 + 9q12 - q14 + 6q16 - 3q20 + 4q22 - 2q24 + q28

HOMFLY-PT Polynomial: a-8 + a-8z2 + a-6z-2 - 2a-6 - 5a-6z2 - 3a-6z4 - 2a-4z-2 - a-4 + 4a-4z2 + 5a-4z4 + 2a-4z6 + a-2z-2 + 2a-2 + a-2z2 + a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial: a-10z2 - 2a-10z4 + a-10z6 - 2a-9z + 6a-9z3 - 8a-9z5 + 4a-9z7 + 3a-8 - 8a-8z2 + 14a-8z4 - 17a-8z6 + 8a-8z8 - 7a-7z + 20a-7z3 - 20a-7z5 - 3a-7z7 + 7a-7z9 + a-6z-2 + 5a-6 - 30a-6z2 + 62a-6z4 - 65a-6z6 + 22a-6z8 + 2a-6z10 - 2a-5z-1 - 5a-5z + 27a-5z3 - 23a-5z5 - 14a-5z7 + 15a-5z9 + 2a-4z-2 + 2a-4 - 27a-4z2 + 63a-4z4 - 67a-4z6 + 25a-4z8 + 2a-4z10 - 2a-3z-1 - a-3z + 19a-3z3 - 22a-3z5 + a-3z7 + 8a-3z9 + a-2z-2 - a-2 - 4a-2z2 + 12a-2z4 - 16a-2z6 + 11a-2z8 - a-1z + 5a-1z3 - 10a-1z5 + 8a-1z7 + 2z2 - 5z4 + 4z6 - az3 + az5

Khovanov Homology:
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8
j = 19           1
j = 17          3 
j = 15         71 
j = 13        103  
j = 11       137   
j = 9      1412    
j = 7     1311     
j = 5    914      
j = 3   813       
j = 1  311        
j = -1 16         
j = -3 3          
j = -51           


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, 486]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 486]]
Out[4]=   
PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[12, 6, 13, 5], 
 
>   X[8, 4, 9, 3], X[22, 14, 19, 13], X[20, 10, 21, 9], X[10, 20, 11, 19], 
 
>   X[14, 22, 15, 21], X[18, 12, 5, 11], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, 5, -3}, {8, -7, 9, -6}, 
 
>   {4, -1, 2, -5, 7, -8, 10, -4, 6, -9, 11, -2, 3, -10}]
In[6]:=
Jones[L][q]
Out[6]=   
      -2   4              2       3       4       5       6       7      8    9
-9 - q   + - + 17 q - 22 q  + 27 q  - 25 q  + 23 q  - 17 q  + 10 q  - 4 q  + q
           q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -6   2     -2      2      4      6      8      10      12    14      16
-2 - q   + -- - q   + 6 q  - 3 q  + 5 q  + 5 q  + 2 q   + 9 q   - q   + 6 q   - 
            4
           q
 
       20      22      24    28
>   3 q   + 4 q   - 2 q   + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 486]][a, z]
Out[8]=   
                                                    2      2      2    2
 -8   2     -4   2      1       2       1      2   z    5 z    4 z    z     4
a   - -- - a   + -- + ----- - ----- + ----- - z  + -- - ---- + ---- + -- - z  - 
       6          2    6  2    4  2    2  2         8     6      4     2
      a          a    a  z    a  z    a  z         a     a      a     a
 
       4      4    4      6    6
    3 z    5 z    z    2 z    z
>   ---- + ---- + -- + ---- + --
      6      4     2     4     2
     a      a     a     a     a
In[9]:=
Kauffman[Link[11, Alternating, 486]][a, z]
Out[9]=   
3    5    2     -2     1       2       1      2      2     2 z   7 z   5 z
-- + -- + -- - a   + ----- + ----- + ----- - ---- - ---- - --- - --- - --- - 
 8    6    4          6  2    4  2    2  2    5      3      9     7     5
a    a    a          a  z    a  z    a  z    a  z   a  z   a     a     a
 
                     2       2       2       2      2      3       3       3
    z    z      2   z     8 z    30 z    27 z    4 z    6 z    20 z    27 z
>   -- - - + 2 z  + --- - ---- - ----- - ----- - ---- + ---- + ----- + ----- + 
     3   a           10     8      6       4       2      9      7       5
    a               a      a      a       a       a      a      a       a
 
        3      3                    4       4       4       4       4      5
    19 z    5 z       3      4   2 z    14 z    62 z    63 z    12 z    8 z
>   ----- + ---- - a z  - 5 z  - ---- + ----- + ----- + ----- + ----- - ---- - 
      3      a                    10      8       6       4       2       9
     a                           a       a       a       a       a       a
 
        5       5       5       5                  6        6       6       6
    20 z    23 z    22 z    10 z       5      6   z     17 z    65 z    67 z
>   ----- - ----- - ----- - ----- + a z  + 4 z  + --- - ----- - ----- - ----- - 
      7       5       3       a                    10     8       6       4
     a       a       a                            a      a       a       a
 
        6      7      7       7    7      7      8       8       8       8
    16 z    4 z    3 z    14 z    z    8 z    8 z    22 z    25 z    11 z
>   ----- + ---- - ---- - ----- + -- + ---- + ---- + ----- + ----- + ----- + 
      2       9      7      5      3    a       8      6       4       2
     a       a      a      a      a            a      a       a       a
 
       9       9      9      10      10
    7 z    15 z    8 z    2 z     2 z
>   ---- + ----- + ---- + ----- + -----
      7      5       3      6       4
     a      a       a      a       a
In[10]:=
Kh[L][q, t]
Out[10]=   
          3     1       3      1      6    3 q       3        5         5  2
11 q + 8 q  + ----- + ----- + ---- + --- + --- + 13 q  t + 9 q  t + 14 q  t  + 
               5  3    3  2      2   q t    t
              q  t    q  t    q t
 
        7  2       7  3       9  3       9  4       11  4      11  5
>   13 q  t  + 11 q  t  + 14 q  t  + 12 q  t  + 13 q   t  + 7 q   t  + 
 
        13  5      13  6      15  6    15  7      17  7    19  8
>   10 q   t  + 3 q   t  + 7 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a486
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