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L11a10

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Acknowledgement

L11a10 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-5/2 - 4q-3/2 + 8q-1/2 - 13q1/2 + 16q3/2 - 20q5/2 + 18q7/2 - 16q9/2 + 12q11/2 - 7q13/2 + 4q15/2 - q17/2

A2 (sl(3)) Invariant: - q-8 + 2q-6 - 3q-2 + 4 - q2 + 2q4 + 5q6 + q8 + 5q10 - q12 + q14 + q16 - 5q18 + q20 - q22 - 2q24 + q26

HOMFLY-PT Polynomial: a-7z-1 - a-7z3 - 2a-5z-1 - 2a-5z + a-5z5 + 2a-3z + 3a-3z3 + 2a-3z5 + a-1z-1 + a-1z5 - az3

Kauffman Polynomial: - 2a-9z3 + 3a-9z5 - a-9z7 + 2a-8 + 3a-8z2 - 16a-8z4 + 15a-8z6 - 4a-8z8 - a-7z-1 + 2a-7z3 - 13a-7z5 + 16a-7z7 - 5a-7z9 + 5a-6 + 3a-6z2 - 32a-6z4 + 32a-6z6 - 5a-6z8 - 2a-6z10 - 2a-5z-1 + 3a-5z - 5a-5z3 - 11a-5z5 + 27a-5z7 - 11a-5z9 + 3a-4 + a-4z2 - 25a-4z4 + 37a-4z6 - 11a-4z8 - 2a-4z10 + 4a-3z - 19a-3z3 + 24a-3z5 - a-3z7 - 6a-3z9 - a-2 - a-2z4 + 12a-2z6 - 10a-2z8 + a-1z-1 + a-1z - 8a-1z3 + 15a-1z5 - 11a-1z7 - z2 + 7z4 - 8z6 + 2az3 - 4az5 - a2z4

Khovanov Homology:
trqj r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8
j = 18           1
j = 16          3 
j = 14         41 
j = 12        83  
j = 10       84   
j = 8      108    
j = 6     108     
j = 4    610      
j = 2   710       
j = 0  38        
j = -2 15         
j = -4 3          
j = -61           


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[11, Alternating, 10]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 10]]
Out[4]=   
PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[12, 6, 13, 5], 
 
>   X[8, 4, 9, 3], X[16, 10, 17, 9], X[22, 14, 5, 13], X[10, 16, 11, 15], 
 
>   X[14, 22, 15, 21], X[20, 12, 21, 11], X[2, 18, 3, 17]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 6, -8, 10, -4, 7, -9, 8, -6, 11, -2, 
 
>    3, -10, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(5/2)    4        8                       3/2       5/2       7/2       9/2
q       - ---- + ------- - 13 Sqrt[q] + 16 q    - 20 q    + 18 q    - 16 q    + 
           3/2   Sqrt[q]
          q
 
        11/2      13/2      15/2    17/2
>   12 q     - 7 q     + 4 q     - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -8   2    3     2      4      6    8      10    12    14    16      18
4 - q   + -- - -- - q  + 2 q  + 5 q  + q  + 5 q   - q   + q   + q   - 5 q   + 
           6    2
          q    q
 
     20    22      24    26
>   q   - q   - 2 q   + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 10]][a, z]
Out[8]=   
                                 3      3           5      5    5
 1      2      1    2 z   2 z   z    3 z       3   z    2 z    z
---- - ---- + --- - --- + --- - -- + ---- - a z  + -- + ---- + --
 7      5     a z    5     3     7     3            5     3    a
a  z   a  z         a     a     a     a            a     a
In[9]:=
Kauffman[Link[11, Alternating, 10]][a, z]
Out[9]=   
                                                                 2      2
2    5    3     -2    1      2      1    3 z   4 z   z    2   3 z    3 z
-- + -- + -- - a   - ---- - ---- + --- + --- + --- + - - z  + ---- + ---- + 
 8    6    4          7      5     a z    5     3    a          8      6
a    a    a          a  z   a  z         a     a               a      a
 
     2      3      3      3       3      3                       4       4
    z    2 z    2 z    5 z    19 z    8 z         3      4   16 z    32 z
>   -- - ---- + ---- - ---- - ----- - ---- + 2 a z  + 7 z  - ----- - ----- - 
     4     9      7      5      3      a                       8       6
    a     a      a      a      a                              a       a
 
        4    4              5       5       5       5       5
    25 z    z     2  4   3 z    13 z    11 z    24 z    15 z         5      6
>   ----- - -- - a  z  + ---- - ----- - ----- + ----- + ----- - 4 a z  - 8 z  + 
      4      2             9      7       5       3       a
     a      a             a      a       a       a
 
        6       6       6       6    7       7       7    7       7      8
    15 z    32 z    37 z    12 z    z    16 z    27 z    z    11 z    4 z
>   ----- + ----- + ----- + ----- - -- + ----- + ----- - -- - ----- - ---- - 
      8       6       4       2      9     7       5      3     a       8
     a       a       a       a      a     a       a      a             a
 
       8       8       8      9       9      9      10      10
    5 z    11 z    10 z    5 z    11 z    6 z    2 z     2 z
>   ---- - ----- - ----- - ---- - ----- - ---- - ----- - -----
      6      4       2       7      5       3      6       4
     a      a       a       a      a       a      a       a
In[10]:=
Kh[L][q, t]
Out[10]=   
       2     1       3       1     3    5         2        4         4  2
8 + 7 q  + ----- + ----- + ----- + - + ---- + 10 q  t + 6 q  t + 10 q  t  + 
            6  3    4  2    2  2   t    2
           q  t    q  t    q  t        q  t
 
        6  2      6  3       8  3      8  4      10  4      10  5      12  5
>   10 q  t  + 8 q  t  + 10 q  t  + 8 q  t  + 8 q   t  + 4 q   t  + 8 q   t  + 
 
       12  6      14  6    14  7      16  7    18  8
>   3 q   t  + 4 q   t  + q   t  + 3 q   t  + q   t


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