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L10n35

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Acknowledgement

L10n35 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - 3q3/2 + 4q5/2 - 8q7/2 + 8q9/2 - 8q11/2 + 8q13/2 - 5q15/2 + 3q17/2 - q19/2

A2 (sl(3)) Invariant: 3q4 + q6 + 5q8 + 5q10 + 2q12 + 3q14 - 3q16 - 2q18 - 4q20 - 3q22 + q24 - q26 + q28 + q30

HOMFLY-PT Polynomial: - a-9z-1 - a-9z + 5a-7z-1 + 7a-7z + 3a-7z3 - 8a-5z-1 - 13a-5z - 8a-5z3 - 2a-5z5 + 4a-3z-1 + 7a-3z + 3a-3z3

Kauffman Polynomial: - a-11z + 2a-11z3 - a-11z5 + 2a-10 - 5a-10z2 + 7a-10z4 - 3a-10z6 - a-9z-1 - a-9z + 4a-9z3 + 3a-9z5 - 3a-9z7 + 9a-8 - 24a-8z2 + 26a-8z4 - 8a-8z6 - a-8z8 - 5a-7z-1 + 9a-7z - 12a-7z3 + 14a-7z5 - 7a-7z7 + 14a-6 - 31a-6z2 + 25a-6z4 - 8a-6z6 - a-6z8 - 8a-5z-1 + 19a-5z - 20a-5z3 + 10a-5z5 - 4a-5z7 + 8a-4 - 12a-4z2 + 6a-4z4 - 3a-4z6 - 4a-3z-1 + 10a-3z - 6a-3z3

Khovanov Homology:
trqj r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8
j = 20        1
j = 18       2 
j = 16      31 
j = 14     52  
j = 12    33   
j = 10   55    
j = 8  33     
j = 6 15      
j = 423       
j = 23        


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[10, NonAlternating, 35]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 35]]
Out[4]=   
PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[13, 17, 14, 16], X[9, 15, 10, 14], 
 
>   X[15, 11, 16, 10], X[17, 5, 18, 20], X[7, 19, 8, 18], X[19, 9, 20, 8], 
 
>   X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, -2, 10}, {9, -1, -7, 8, -4, 5, -10, 2, -3, 4, -5, 3, -6, 7, 
 
>    -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
    3/2      5/2      7/2      9/2      11/2      13/2      15/2      17/2
-3 q    + 4 q    - 8 q    + 8 q    - 8 q     + 8 q     - 5 q     + 3 q     - 
 
     19/2
>   q
In[7]:=
A2Invariant[L][q]
Out[7]=   
   4    6      8      10      12      14      16      18      20      22
3 q  + q  + 5 q  + 5 q   + 2 q   + 3 q   - 3 q   - 2 q   - 4 q   - 3 q   + 
 
     24    26    28    30
>   q   - q   + q   + q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 35]][a, z]
Out[8]=   
                                                          3      3      3      5
   1       5      8      4     z    7 z   13 z   7 z   3 z    8 z    3 z    2 z
-(----) + ---- - ---- + ---- - -- + --- - ---- + --- + ---- - ---- + ---- - ----
   9       7      5      3      9    7      5     3      7      5      3      5
  a  z    a  z   a  z   a  z   a    a      a     a      a      a      a      a
In[9]:=
Kauffman[Link[10, NonAlternating, 35]][a, z]
Out[9]=   
 2    9    14   8     1      5      8      4      z    z    9 z   19 z   10 z
--- + -- + -- + -- - ---- - ---- - ---- - ---- - --- - -- + --- + ---- + ---- - 
 10    8    6    4    9      7      5      3      11    9    7      5      3
a     a    a    a    a  z   a  z   a  z   a  z   a     a    a      a      a
 
       2       2       2       2      3      3       3       3      3      4
    5 z    24 z    31 z    12 z    2 z    4 z    12 z    20 z    6 z    7 z
>   ---- - ----- - ----- - ----- + ---- + ---- - ----- - ----- - ---- + ---- + 
     10      8       6       4      11      9      7       5       3     10
    a       a       a       a      a       a      a       a       a     a
 
        4       4      4    5       5       5       5      6      6      6
    26 z    25 z    6 z    z     3 z    14 z    10 z    3 z    8 z    8 z
>   ----- + ----- + ---- - --- + ---- + ----- + ----- - ---- - ---- - ---- - 
      8       6       4     11     9      7       5      10      8      6
     a       a       a     a      a      a       a      a       a      a
 
       6      7      7      7    8    8
    3 z    3 z    7 z    4 z    z    z
>   ---- - ---- - ---- - ---- - -- - --
      4      9      7      5     8    6
     a      a      a      a     a    a
In[10]:=
Kh[L][q, t]
Out[10]=   
   2      4      4      6        6  2      8  2      8  3      10  3
3 q  + 2 q  + 3 q  t + q  t + 5 q  t  + 3 q  t  + 3 q  t  + 5 q   t  + 
 
       10  4      12  4      12  5      14  5      14  6      16  6    16  7
>   5 q   t  + 3 q   t  + 3 q   t  + 5 q   t  + 2 q   t  + 3 q   t  + q   t  + 
 
       18  7    20  8
>   2 q   t  + q   t


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