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L11a440

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Acknowledgement

L11a440 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-11 + 3q-10 - 6q-9 + 10q-8 - 13q-7 + 17q-6 - 15q-5 + 14q-4 - 10q-3 + 7q-2 - 3q-1 + 1

A2 (sl(3)) Invariant: - q-34 - q-32 + q-30 - q-28 + 3q-26 + 3q-24 + q-22 + 6q-20 + q-18 + 5q-16 + 3q-14 + q-12 + 5q-10 - q-8 + 2q-6 - q-2 + 1

HOMFLY-PT Polynomial: 2a2z2 + a2z4 + a4z-2 + 5a4 + 5a4z2 - a4z4 - a4z6 - 2a6z-2 - 9a6 - 12a6z2 - 8a6z4 - 2a6z6 + a8z-2 + 6a8 + 9a8z2 + 3a8z4 - 2a10 - a10z2

Kauffman Polynomial: 2a2z2 - 3a2z4 + a2z6 + 4a3z3 - 8a3z5 + 3a3z7 - a4z-2 + 6a4 - 14a4z2 + 16a4z4 - 15a4z6 + 5a4z8 + 2a5z-1 - 5a5z + 3a5z3 + 2a5z5 - 8a5z7 + 4a5z9 - 2a6z-2 + 13a6 - 38a6z2 + 54a6z4 - 37a6z6 + 9a6z8 + a6z10 + 2a7z-1 - 8a7z + 7a7z3 + 9a7z5 - 15a7z7 + 7a7z9 - a8z-2 + 9a8 - 23a8z2 + 38a8z4 - 27a8z6 + 8a8z8 + a8z10 - 3a9z + 9a9z3 - 7a9z5 + 3a9z9 + 2a10z2 - 3a10z4 - 3a10z6 + 4a10z8 + a11z - a11z3 - 5a11z5 + 4a11z7 - a12 + 3a12z2 - 6a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 1           1
j = -1          2 
j = -3         51 
j = -5        63  
j = -7       84   
j = -9      76    
j = -11     108     
j = -13    610      
j = -15   47       
j = -17  26        
j = -19 14         
j = -21 2          
j = -231           


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, Alternating, 440]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 440]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[12, 15, 5, 16], X[8, 17, 9, 18], 
 
>   X[16, 7, 17, 8], X[18, 9, 19, 10], X[22, 19, 13, 20], X[20, 12, 21, 11], 
 
>   X[10, 22, 11, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 5, -4, 6, -9, 8, -3}, 
 
>   {11, -2, 3, -5, 4, -6, 7, -8, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
     -11    3    6    10   13   17   15   14   10   7    3
1 - q    + --- - -- + -- - -- + -- - -- + -- - -- + -- - -
            10    9    8    7    6    5    4    3    2   q
           q     q    q    q    q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -34    -32    -30    -28    3     3     -22    6     -18    5     3
1 - q    - q    + q    - q    + --- + --- + q    + --- + q    + --- + --- + 
                                 26    24           20           16    14
                                q     q            q            q     q
 
     -12    5     -8   2     -2
>   q    + --- - q   + -- - q
            10          6
           q           q
In[8]:=
HOMFLYPT[Link[11, Alternating, 440]][a, z]
Out[8]=   
                              4      6    8
   4      6      8      10   a    2 a    a       2  2      4  2       6  2
5 a  - 9 a  + 6 a  - 2 a   + -- - ---- + -- + 2 a  z  + 5 a  z  - 12 a  z  + 
                              2     2     2
                             z     z     z
 
       8  2    10  2    2  4    4  4      6  4      8  4    4  6      6  6
>   9 a  z  - a   z  + a  z  - a  z  - 8 a  z  + 3 a  z  - a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 440]][a, z]
Out[9]=   
                             4      6    8      5      7
   4       6      8    12   a    2 a    a    2 a    2 a       5        7
6 a  + 13 a  + 9 a  - a   - -- - ---- - -- + ---- + ---- - 5 a  z - 8 a  z - 
                             2     2     2    z      z
                            z     z     z
 
       9      11      13        2  2       4  2       6  2       8  2
>   3 a  z + a   z + a   z + 2 a  z  - 14 a  z  - 38 a  z  - 23 a  z  + 
 
       10  2      12  2      3  3      5  3      7  3      9  3    11  3
>   2 a   z  + 3 a   z  + 4 a  z  + 3 a  z  + 7 a  z  + 9 a  z  - a   z  - 
 
       13  3      2  4       4  4       6  4       8  4      10  4      12  4
>   2 a   z  - 3 a  z  + 16 a  z  + 54 a  z  + 38 a  z  - 3 a   z  - 6 a   z  - 
 
       3  5      5  5      7  5      9  5      11  5    13  5    2  6
>   8 a  z  + 2 a  z  + 9 a  z  - 7 a  z  - 5 a   z  + a   z  + a  z  - 
 
        4  6       6  6       8  6      10  6      12  6      3  7      5  7
>   15 a  z  - 37 a  z  - 27 a  z  - 3 a   z  + 3 a   z  + 3 a  z  - 8 a  z  - 
 
        7  7      11  7      4  8      6  8      8  8      10  8      5  9
>   15 a  z  + 4 a   z  + 5 a  z  + 9 a  z  + 8 a  z  + 4 a   z  + 4 a  z  + 
 
       7  9      9  9    6  10    8  10
>   7 a  z  + 3 a  z  + a  z   + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3    5      1        2        1        4        2        6        4
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 5    3    23  9    21  8    19  8    19  7    17  7    17  6    15  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      7        6        10       10       8        7       6       8      4
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 
     15  5    13  5    13  4    11  4    11  3    9  3    9  2    7  2    7
    q   t    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
     6     t    2 t      2
>   ---- + -- + --- + q t
     5      3    q
    q  t   q


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