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L11n377

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Acknowledgement

L11n377 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X17,22,18,19 X11,20,12,21 X19,12,20,13 X21,18,22,5 X9,16,10,17 X2,14,3,13

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

Jones Polynomial: - q-10 + 2q-9 - 4q-8 + 6q-7 - 7q-6 + 9q-5 - 6q-4 + 7q-3 - 4q-2 + 2q-1

A2 (sl(3)) Invariant: - q-32 - 2q-30 - q-28 - 3q-26 - q-24 + 3q-22 + 2q-20 + 6q-18 + 6q-16 + 6q-14 + 6q-12 + 2q-10 + 3q-8 - q-4 + 2q-2

HOMFLY-PT Polynomial: a2 + 2a2z2 + 2a4z-2 + 4a4 + a4z2 - a4z4 - 5a6z-2 - 10a6 - 6a6z2 - 2a6z4 + 4a8z-2 + 6a8 + 3a8z2 - a10z-2 - a10

Kauffman Polynomial: - a2 + 3a2z2 + 3a3z3 + a3z5 - 2a4z-2 + 6a4 - 5a4z2 - a4z4 + 3a4z6 + 5a5z-1 - 15a5z + 19a5z3 - 14a5z5 + 5a5z7 - 5a6z-2 + 16a6 - 25a6z2 + 18a6z4 - 12a6z6 + 4a6z8 + 9a7z-1 - 30a7z + 41a7z3 - 29a7z5 + 5a7z7 + a7z9 - 4a8z-2 + 13a8 - 23a8z2 + 31a8z4 - 24a8z6 + 6a8z8 + 5a9z-1 - 20a9z + 33a9z3 - 19a9z5 + a9z7 + a9z9 - a10z-2 + 3a10 - 6a10z2 + 12a10z4 - 9a10z6 + 2a10z8 + a11z-1 - 5a11z + 8a11z3 - 5a11z5 + a11z7

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -1         2
j = -3        42
j = -5       3  
j = -7      34  
j = -9     63   
j = -11    35    
j = -13   34     
j = -15  13      
j = -17 13       
j = -19 1        
j = -211         


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, 377]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 377]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[15, 1, 16, 4], X[5, 10, 6, 11], 
 
>   X[3, 8, 4, 9], X[17, 22, 18, 19], X[11, 20, 12, 21], X[19, 12, 20, 13], 
 
>   X[21, 18, 22, 5], X[9, 16, 10, 17], X[2, 14, 3, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, -5, 3}, {-8, 7, -9, 6}, 
 
>   {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, -6, 9}]
In[6]:=
Jones[L][q]
Out[6]=   
  -10   2    4    6    7    9    6    7    4    2
-q    + -- - -- + -- - -- + -- - -- + -- - -- + -
         9    8    7    6    5    4    3    2   q
        q    q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -32    2     -28    3     -24    3     2     6     6     6     6     2
-q    - --- - q    - --- - q    + --- + --- + --- + --- + --- + --- + --- + 
         30           26           22    20    18    16    14    12    10
        q            q            q     q     q     q     q     q     q
 
    3     -4   2
>   -- - q   + --
     8          2
    q          q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 377]][a, z]
Out[8]=   
                                    4      6      8    10
 2      4       6      8    10   2 a    5 a    4 a    a        2  2    4  2
a  + 4 a  - 10 a  + 6 a  - a   + ---- - ---- + ---- - --- + 2 a  z  + a  z  - 
                                   2      2      2     2
                                  z      z      z     z
 
       6  2      8  2    4  4      6  4
>   6 a  z  + 3 a  z  - a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 377]][a, z]
Out[9]=   
                                        4      6      8    10      5      7
  2      4       6       8      10   2 a    5 a    4 a    a     5 a    9 a
-a  + 6 a  + 16 a  + 13 a  + 3 a   - ---- - ---- - ---- - --- + ---- + ---- + 
                                       2      2      2     2     z      z
                                      z      z      z     z
 
       9    11
    5 a    a         5         7         9        11        2  2      4  2
>   ---- + --- - 15 a  z - 30 a  z - 20 a  z - 5 a   z + 3 a  z  - 5 a  z  - 
     z      z
 
        6  2       8  2      10  2      3  3       5  3       7  3       9  3
>   25 a  z  - 23 a  z  - 6 a   z  + 3 a  z  + 19 a  z  + 41 a  z  + 33 a  z  + 
 
       11  3    4  4       6  4       8  4       10  4    3  5       5  5
>   8 a   z  - a  z  + 18 a  z  + 31 a  z  + 12 a   z  + a  z  - 14 a  z  - 
 
        7  5       9  5      11  5      4  6       6  6       8  6      10  6
>   29 a  z  - 19 a  z  - 5 a   z  + 3 a  z  - 12 a  z  - 24 a  z  - 9 a   z  + 
 
       5  7      7  7    9  7    11  7      6  8      8  8      10  8    7  9
>   5 a  z  + 5 a  z  + a  z  + a   z  + 4 a  z  + 6 a  z  + 2 a   z  + a  z  + 
 
     9  9
>   a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
2    2     1        1        1        3        1        3        3
-- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 3   q    21  9    19  8    17  8    17  7    15  7    15  6    13  6
q        q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      4        3        5        6       3       3       4       3      4
>   ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----
     13  5    11  5    11  4    9  4    9  3    7  3    7  2    5  2    3
    q   t    q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t


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