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L11n265

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Acknowledgement

L11n265 as Morse Link
DrawMorseLink

PD Presentation: X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X9,1,10,4

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

Jones Polynomial: 3q2 - 4q3 + 10q4 - 10q5 + 12q6 - 11q7 + 9q8 - 6q9 + 2q10 - q11

A2 (sl(3)) Invariant: 3q6 + q8 + 7q10 + 7q12 + 7q14 + 10q16 + 3q18 + 5q20 - q22 - 2q24 - 2q26 - 6q28 - 2q30 - 2q32 - q34

HOMFLY-PT Polynomial: - 2a-10z-2 - 3a-10 - a-10z2 + 7a-8z-2 + 15a-8 + 11a-8z2 + 3a-8z4 - 8a-6z-2 - 22a-6 - 21a-6z2 - 10a-6z4 - 2a-6z6 + 3a-4z-2 + 10a-4 + 10a-4z2 + 3a-4z4

Kauffman Polynomial: - a-13z-1 + 3a-13z - 3a-13z3 + a-13z5 + a-12 - 3a-12z4 + 2a-12z6 - a-11z-1 + 3a-11z - 2a-11z3 - 3a-11z5 + 3a-11z7 - 2a-10z-2 + 7a-10 - 10a-10z2 + 4a-10z4 - 3a-10z6 + 3a-10z8 + 7a-9z-1 - 21a-9z + 28a-9z3 - 21a-9z5 + 7a-9z7 + a-9z9 - 7a-8z-2 + 22a-8 - 35a-8z2 + 32a-8z4 - 18a-8z6 + 7a-8z8 + 15a-7z-1 - 45a-7z + 48a-7z3 - 26a-7z5 + 7a-7z7 + a-7z9 - 8a-6z-2 + 28a-6 - 41a-6z2 + 31a-6z4 - 13a-6z6 + 4a-6z8 + 8a-5z-1 - 24a-5z + 21a-5z3 - 9a-5z5 + 3a-5z7 - 3a-4z-2 + 13a-4 - 16a-4z2 + 6a-4z4

Khovanov Homology:
trqj r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 23         1
j = 21        1 
j = 19       51 
j = 17      41  
j = 15     75   
j = 13    54    
j = 11   68     
j = 9  44      
j = 7  6       
j = 534        
j = 33         


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, NonAlternating, 265]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 265]]
Out[4]=   
PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], 
 
>   X[11, 19, 12, 18], X[15, 21, 16, 20], X[17, 9, 18, 22], X[21, 17, 22, 16], 
 
>   X[19, 13, 20, 12], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, -2, 11}, {10, -1, -3, 4}, 
 
>   {-11, 2, -5, 9, -4, 3, -6, 8, -7, 5, -9, 6, -8, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
   2      3       4       5       6       7      8      9      10    11
3 q  - 4 q  + 10 q  - 10 q  + 12 q  - 11 q  + 9 q  - 6 q  + 2 q   - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
   6    8      10      12      14       16      18      20    22      24
3 q  + q  + 7 q   + 7 q   + 7 q   + 10 q   + 3 q   + 5 q   - q   - 2 q   - 
 
       26      28      30      32    34
>   2 q   - 6 q   - 2 q   - 2 q   - q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 265]][a, z]
Out[8]=   
                                                       2        2       2
-3    15   22   10     2        7       8       3     z     11 z    21 z
--- + -- - -- + -- - ------ + ----- - ----- + ----- - --- + ----- - ----- + 
 10    8    6    4    10  2    8  2    6  2    4  2    10     8       6
a     a    a    a    a   z    a  z    a  z    a  z    a      a       a
 
        2      4       4      4      6
    10 z    3 z    10 z    3 z    2 z
>   ----- + ---- - ----- + ---- - ----
      4       8      6       4      6
     a       a      a       a      a
In[9]:=
Kauffman[Link[11, NonAlternating, 265]][a, z]
Out[9]=   
 -12    7    22   28   13     2        7       8       3       1       1
a    + --- + -- + -- + -- - ------ - ----- - ----- - ----- - ----- - ----- + 
        10    8    6    4    10  2    8  2    6  2    4  2    13      11
       a     a    a    a    a   z    a  z    a  z    a  z    a   z   a   z
 
                                                              2       2
     7      15     8     3 z   3 z   21 z   45 z   24 z   10 z    35 z
>   ---- + ---- + ---- + --- + --- - ---- - ---- - ---- - ----- - ----- - 
     9      7      5      13    11     9      7      5      10      8
    a  z   a  z   a  z   a     a      a      a      a      a       a
 
        2       2      3      3       3       3       3      4      4       4
    41 z    16 z    3 z    2 z    28 z    48 z    21 z    3 z    4 z    32 z
>   ----- - ----- - ---- - ---- + ----- + ----- + ----- - ---- + ---- + ----- + 
      6       4      13     11      9       7       5      12     10      8
     a       a      a      a       a       a       a      a      a       a
 
        4      4    5       5       5       5      5      6      6       6
    31 z    6 z    z     3 z    21 z    26 z    9 z    2 z    3 z    18 z
>   ----- + ---- + --- - ---- - ----- - ----- - ---- + ---- - ---- - ----- - 
      6       4     13    11      9       7       5     12     10      8
     a       a     a     a       a       a       a     a      a       a
 
        6      7      7      7      7      8      8      8    9    9
    13 z    3 z    7 z    7 z    3 z    3 z    7 z    4 z    z    z
>   ----- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + --
      6      11      9      7      5     10      8      6     9    7
     a      a       a      a      a     a       a      a     a    a
In[10]:=
Kh[L][q, t]
Out[10]=   
   3      5      5        7  2      9  2      9  3      11  3      11  4
3 q  + 3 q  + 4 q  t + 6 q  t  + 4 q  t  + 4 q  t  + 6 q   t  + 8 q   t  + 
 
       13  4      13  5      15  5      15  6      17  6    17  7      19  7
>   5 q   t  + 4 q   t  + 7 q   t  + 5 q   t  + 4 q   t  + q   t  + 5 q   t  + 
 
     19  8    21  8    23  9
>   q   t  + q   t  + q   t


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