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L11a160

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Acknowledgement

L11a160 as Morse Link
DrawMorseLink

PD Presentation: X8192 X10,4,11,3 X22,10,7,9 X2738 X16,14,17,13 X14,5,15,6 X4,15,5,16 X20,17,21,18 X18,11,19,12 X12,19,13,20 X6,22,1,21

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

Jones Polynomial: q-15/2 - 3q-13/2 + 7q-11/2 - 12q-9/2 + 16q-7/2 - 20q-5/2 + 19q-3/2 - 18q-1/2 + 13q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant: - q-24 - q-22 + q-20 - 2q-18 + 3q-14 - 2q-12 + 5q-10 + 2q-8 + 2q-6 + 4q-4 - 3q-2 + 4 - 3q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial: - a-1z - 2a-1z3 - a-1z5 + 4az + 7az3 + 4az5 + az7 - 2a3z-1 - 9a3z - 9a3z3 - 3a3z5 + 3a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial: a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + 2a-1z - 6a-1z3 + 11a-1z5 - 7a-1z7 - 4z2 + 4z4 + 6z6 - 7z8 + 8az - 24az3 + 26az5 - 7az7 - 4az9 - 2a2z2 - 11a2z4 + 25a2z6 - 13a2z8 - a2z10 - 2a3z-1 + 15a3z - 33a3z3 + 27a3z5 - 7a3z9 + 3a4 - 4a4z2 - 9a4z4 + 22a4z6 - 10a4z8 - a4z10 - 3a5z-1 + 12a5z - 23a5z3 + 21a5z5 - 3a5z7 - 3a5z9 + 3a6 - 7a6z2 + 3a6z4 + 6a6z6 - 4a6z8 - a7z-1 + 3a7z - 7a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8           1
j = 6          3 
j = 4         51 
j = 2        83  
j = 0       105   
j = -2      109    
j = -4     109     
j = -6    711      
j = -8   59       
j = -10  27        
j = -12 15         
j = -14 2          
j = -161           


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, 160]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 160]]
Out[4]=   
PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], 
 
>   X[16, 14, 17, 13], X[14, 5, 15, 6], X[4, 15, 5, 16], X[20, 17, 21, 18], 
 
>   X[18, 11, 19, 12], X[12, 19, 13, 20], X[6, 22, 1, 21]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, 2, -7, 6, -11}, 
 
>   {4, -1, 3, -2, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8, 11, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     3       7      12     16     20     19      18
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
 
       3/2      5/2    7/2
>   8 q    + 4 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    -22    -20    2     3     2     5    2    2    4    3       2
4 - q    - q    + q    - --- + --- - --- + --- + -- + -- + -- - -- - 3 q  - 
                          18    14    12    10    8    6    4    2
                         q     q     q     q     q    q    q    q
 
     4      6      8    10
>   q  + 2 q  - 2 q  + q
In[8]:=
HOMFLYPT[Link[11, Alternating, 160]][a, z]
Out[8]=   
    3      5    7                                           3
-2 a    3 a    a    z              3        5      7     2 z         3
----- + ---- - -- - - + 4 a z - 9 a  z + 6 a  z - a  z - ---- + 7 a z  - 
  z      z     z    a                                     a
 
                         5
       3  3      5  3   z         5      3  5      7
>   9 a  z  + 3 a  z  - -- + 4 a z  - 3 a  z  + a z
                        a
In[9]:=
Kauffman[Link[11, Alternating, 160]][a, z]
Out[9]=   
                      3      5    7
   4      6    8   2 a    3 a    a    2 z               3         5
3 a  + 3 a  + a  - ---- - ---- - -- + --- + 8 a z + 15 a  z + 12 a  z + 
                    z      z     z     a
 
                       2                                            3      3
       7        2   2 z       2  2      4  2      6  2      8  2   z    6 z
>   3 a  z - 4 z  - ---- - 2 a  z  - 4 a  z  - 7 a  z  - 3 a  z  + -- - ---- - 
                      2                                             3    a
                     a                                             a
 
                                                        4
          3       3  3       5  3      7  3      4   6 z        2  4
>   24 a z  - 33 a  z  - 23 a  z  - 7 a  z  + 4 z  + ---- - 11 a  z  - 
                                                       2
                                                      a
 
                                   5       5
       4  4      6  4      8  4   z    11 z          5       3  5       5  5
>   9 a  z  + 3 a  z  + 3 a  z  - -- + ----- + 26 a z  + 27 a  z  + 21 a  z  + 
                                   3     a
                                  a
 
                        6                                              7
       7  5      6   4 z        2  6       4  6      6  6    8  6   7 z
>   8 a  z  + 6 z  - ---- + 25 a  z  + 22 a  z  + 6 a  z  - a  z  - ---- - 
                       2                                             a
                      a
 
         7      5  7      7  7      8       2  8       4  8      6  8
>   7 a z  - 3 a  z  - 3 a  z  - 7 z  - 13 a  z  - 10 a  z  - 4 a  z  - 
 
         9      3  9      5  9    2  10    4  10
>   4 a z  - 7 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
     9      1        2        1        5        2        7        5       9
10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
      2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
     q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      7      11      10      9      10             2        2  2      4  2
>   ----- + ----- + ----- + ---- + ---- + 5 t + 8 q  t + 3 q  t  + 5 q  t  + 
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t
 
     4  3      6  3    8  4
>   q  t  + 3 q  t  + q  t


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