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L11a70

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Acknowledgement

L11a70 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X16,8,17,7 X22,18,5,17 X18,9,19,10 X8,21,9,22 X10,14,11,13 X20,15,21,16 X14,19,15,20 X2536 X4,11,1,12

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

Jones Polynomial: - q-19/2 + 3q-17/2 - 7q-15/2 + 13q-13/2 - 19q-11/2 + 22q-9/2 - 23q-7/2 + 20q-5/2 - 16q-3/2 + 10q-1/2 - 5q1/2 + q3/2

A2 (sl(3)) Invariant: q-30 + q-28 - q-26 + 2q-24 - 4q-20 + 3q-18 - 3q-16 + 2q-14 + 3q-12 - q-10 + 6q-8 - 4q-6 + 3q-4 + q-2 - 2 + 3q2 - q4

HOMFLY-PT Polynomial: - az + az3 + az5 - 2a3z-1 - 4a3z - 4a3z3 - 3a3z5 - a3z7 + 4a5z-1 + 8a5z + 8a5z3 + 3a5z5 - 3a7z-1 - 6a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial: z4 - z6 - az - 4az3 + 10az5 - 5az7 + a2z2 - 10a2z4 + 20a2z6 - 9a2z8 - 2a3z-1 + 7a3z - 15a3z3 + 15a3z5 + 6a3z7 - 7a3z9 + 2a4 + 2a4z2 - 26a4z4 + 42a4z6 - 15a4z8 - 2a4z10 - 4a5z-1 + 18a5z - 30a5z3 + 15a5z5 + 14a5z7 - 12a5z9 + 3a6 - 4a6z2 - 16a6z4 + 28a6z6 - 12a6z8 - 2a6z10 - 3a7z-1 + 13a7z - 24a7z3 + 17a7z5 - 2a7z7 - 5a7z9 + 3a8 - 8a8z2 + 4a8z4 + 4a8z6 - 6a8z8 - a9z-1 + 2a9z - 3a9z3 + 6a9z5 - 5a9z7 + a10 - 3a10z2 + 5a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 4           1
j = 2          4 
j = 0         61 
j = -2        104  
j = -4       117   
j = -6      129    
j = -8     1011     
j = -10    912      
j = -12   511       
j = -14  28        
j = -16 15         
j = -18 2          
j = -201           


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, 70]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 70]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], 
 
>   X[18, 9, 19, 10], X[8, 21, 9, 22], X[10, 14, 11, 13], X[20, 15, 21, 16], 
 
>   X[14, 19, 15, 20], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -7, 11, -2, 7, -9, 8, -3, 4, -5, 
 
>    9, -8, 6, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(19/2)     3       7      13      19      22     23     20     16
-q        + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 
             17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
            q       q       q       q       q      q      q      q
 
      10                   3/2
>   ------- - 5 Sqrt[q] + q
    Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -30    -28    -26    2     4     3     3     2     3     -10   6    4
-2 + q    + q    - q    + --- - --- + --- - --- + --- + --- - q    + -- - -- + 
                           24    20    18    16    14    12           8    6
                          q     q     q     q     q     q            q    q
 
    3     -2      2    4
>   -- + q   + 3 q  - q
     4
    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 70]][a, z]
Out[8]=   
    3      5      7    9
-2 a    4 a    3 a    a             3        5        7      9        3
----- + ---- - ---- + -- - a z - 4 a  z + 8 a  z - 6 a  z + a  z + a z  - 
  z      z      z     z
 
       3  3      5  3      7  3      5      3  5      5  5    3  7
>   4 a  z  + 8 a  z  - 3 a  z  + a z  - 3 a  z  + 3 a  z  - a  z
In[9]:=
Kauffman[Link[11, Alternating, 70]][a, z]
Out[9]=   
                              3      5      7    9
   4      6      8    10   2 a    4 a    3 a    a             3         5
2 a  + 3 a  + 3 a  + a   - ---- - ---- - ---- - -- - a z + 7 a  z + 18 a  z + 
                            z      z      z     z
 
        7        9      11      2  2      4  2      6  2      8  2      10  2
>   13 a  z + 2 a  z - a   z + a  z  + 2 a  z  - 4 a  z  - 8 a  z  - 3 a   z  - 
 
         3       3  3       5  3       7  3      9  3      11  3    4
>   4 a z  - 15 a  z  - 30 a  z  - 24 a  z  - 3 a  z  + 2 a   z  + z  - 
 
        2  4       4  4       6  4      8  4      10  4         5       3  5
>   10 a  z  - 26 a  z  - 16 a  z  + 4 a  z  + 5 a   z  + 10 a z  + 15 a  z  + 
 
        5  5       7  5      9  5    11  5    6       2  6       4  6
>   15 a  z  + 17 a  z  + 6 a  z  - a   z  - z  + 20 a  z  + 42 a  z  + 
 
        6  6      8  6      10  6        7      3  7       5  7      7  7
>   28 a  z  + 4 a  z  - 3 a   z  - 5 a z  + 6 a  z  + 14 a  z  - 2 a  z  - 
 
       9  7      2  8       4  8       6  8      8  8      3  9       5  9
>   5 a  z  - 9 a  z  - 15 a  z  - 12 a  z  - 6 a  z  - 7 a  z  - 12 a  z  - 
 
       7  9      4  10      6  10
>   5 a  z  - 2 a  z   - 2 a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
7    10     1        2        1        5        2        8        5
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    20  8    18  7    16  7    16  6    14  6    14  5    12  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      11       9        12      10      11      12      9      11
>   ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 t + 
     12  4    10  4    10  3    8  3    8  2    6  2    6      4
    q   t    q   t    q   t    q  t    q  t    q  t    q  t   q  t
 
    4 t    2      2  2    4  3
>   --- + t  + 4 q  t  + q  t
     2
    q


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