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L11a415

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Acknowledgement

L11a415 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-3 - 2q-2 + 6q-1 - 9 + 14q - 15q2 + 17q3 - 14q4 + 11q5 - 7q6 + 3q7 - q8

A2 (sl(3)) Invariant: q-10 + q-8 + 3q-4 + 2 + 7q2 + 3q4 + 9q6 + 4q8 + 3q10 + 2q12 - 4q14 + q16 - 3q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial: - a-6z-2 - 2a-6 - 2a-6z2 - a-6z4 + 4a-4z-2 + 8a-4 + 6a-4z2 + 3a-4z4 + a-4z6 - 5a-2z-2 - 8a-2 - 4a-2z2 + a-2z4 + a-2z6 + 2z-2 - 4z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial: - 2a-9z3 + a-9z5 - 5a-8z4 + 3a-8z6 + a-7z-1 - 5a-7z + 12a-7z3 - 14a-7z5 + 6a-7z7 - a-6z-2 + 4a-6 - 12a-6z2 + 22a-6z4 - 18a-6z6 + 7a-6z8 + 5a-5z-1 - 21a-5z + 30a-5z3 - 13a-5z5 - 2a-5z7 + 4a-5z9 - 4a-4z-2 + 17a-4 - 33a-4z2 + 44a-4z4 - 30a-4z6 + 9a-4z8 + a-4z10 + 9a-3z-1 - 33a-3z + 35a-3z3 - 9a-3z5 - 8a-3z7 + 6a-3z9 - 5a-2z-2 + 20a-2 - 33a-2z2 + 27a-2z4 - 17a-2z6 + 5a-2z8 + a-2z10 + 5a-1z-1 - 16a-1z + 21a-1z3 - 16a-1z5 + 2a-1z7 + 2a-1z9 - 2z-2 + 6 - 7z2 + 6z4 - 7z6 + 3z8 + az + 2az3 - 5az5 + 2az7 - 2a2 + 5a2z2 - 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        62  
j = 9       85   
j = 7      96    
j = 5     810     
j = 3    67      
j = 1   49       
j = -1  25        
j = -3  4         
j = -512          
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, 415]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 415]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 10, 19, 9], X[16, 8, 17, 7], 
 
>   X[22, 16, 11, 15], X[20, 14, 21, 13], X[14, 22, 15, 21], X[10, 18, 5, 17], 
 
>   X[8, 20, 9, 19], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, 
 
>   {11, -2, 6, -7, 5, -4, 8, -3, 9, -6, 7, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
      -3   2    6              2       3       4       5      6      7    8
-9 + q   - -- + - + 14 q - 15 q  + 17 q  - 14 q  + 11 q  - 7 q  + 3 q  - q
            2   q
           q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -10    -8   3       2      4      6      8      10      12      14    16
2 + q    + q   + -- + 7 q  + 3 q  + 9 q  + 4 q  + 3 q   + 2 q   - 4 q   + q   - 
                  4
                 q
 
       18      20    22    24
>   3 q   - 2 q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 415]][a, z]
Out[8]=   
                                                             2      2      2
-2   8    8       2   2      1       4       5        2   2 z    6 z    4 z
-- + -- - -- + 2 a  + -- - ----- + ----- - ----- - 4 z  - ---- + ---- - ---- + 
 6    4    2           2    6  2    4  2    2  2            6      4      2
a    a    a           z    a  z    a  z    a  z            a      a      a
 
                    4      4    4    6    6
     2  2      4   z    3 z    z    z    z
>   a  z  - 2 z  - -- + ---- + -- + -- + --
                    6     4     2    4    2
                   a     a     a    a    a
In[9]:=
Kauffman[Link[11, Alternating, 415]][a, z]
Out[9]=   
    4    17   20      2   2      1       4       5      1      5      9
6 + -- + -- + -- - 2 a  - -- - ----- - ----- - ----- + ---- + ---- + ---- + 
     6    4    2           2    6  2    4  2    2  2    7      5      3
    a    a    a           z    a  z    a  z    a  z    a  z   a  z   a  z
 
                                                      2       2       2
     5    5 z   21 z   33 z   16 z            2   12 z    33 z    33 z
>   --- - --- - ---- - ---- - ---- + a z - 7 z  - ----- - ----- - ----- + 
    a z    7      5      3     a                    6       4       2
          a      a      a                          a       a       a
 
                 3       3       3       3       3                      4
       2  2   2 z    12 z    30 z    35 z    21 z         3      4   5 z
>   5 a  z  - ---- + ----- + ----- + ----- + ----- + 2 a z  + 6 z  - ---- + 
                9      7       5       3       a                       8
               a      a       a       a                               a
 
        4       4       4              5       5       5      5       5
    22 z    44 z    27 z       2  4   z    14 z    13 z    9 z    16 z
>   ----- + ----- + ----- - 4 a  z  + -- - ----- - ----- - ---- - ----- - 
      6       4       2                9     7       5       3      a
     a       a       a                a     a       a       a
 
                       6       6       6       6              7      7      7
         5      6   3 z    18 z    30 z    17 z     2  6   6 z    2 z    8 z
>   5 a z  - 7 z  + ---- - ----- - ----- - ----- + a  z  + ---- - ---- - ---- + 
                      8      6       4       2               7      5      3
                     a      a       a       a               a      a      a
 
       7                      8      8      8      9      9      9    10    10
    2 z         7      8   7 z    9 z    5 z    4 z    6 z    2 z    z     z
>   ---- + 2 a z  + 3 z  + ---- + ---- + ---- + ---- + ---- + ---- + --- + ---
     a                       6      4      2      5      3     a      4     2
                            a      a      a      a      a            a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
         3     1       1       2       4      2      5    4 q      3
9 q + 6 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 q  t + 
              7  4    5  4    5  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
>   8 q  t + 10 q  t  + 9 q  t  + 6 q  t  + 8 q  t  + 5 q  t  + 6 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


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