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L11a62

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Acknowledgement

L11a62 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-21/2 - 3q-19/2 + 7q-17/2 - 12q-15/2 + 16q-13/2 - 20q-11/2 + 20q-9/2 - 18q-7/2 + 13q-5/2 - 9q-3/2 + 4q-1/2 - q1/2

A2 (sl(3)) Invariant: - q-34 - 2q-32 + q-30 - 2q-26 + 5q-24 + 4q-18 - 2q-16 + 3q-14 - 2q-12 + q-10 + 4q-8 - 3q-6 + 4q-4 - 2 + q2

HOMFLY-PT Polynomial: - az3 - a3z-1 - 3a3z - a3z3 + a3z5 + 2a5z-1 + 4a5z + 3a5z3 + 2a5z5 - 3a7z-1 - 7a7z - 5a7z3 + 3a9z-1 + 4a9z - a11z-1

Kauffman Polynomial: az3 - az5 + 5a2z4 - 4a2z6 - a3z-1 + 4a3z - 8a3z3 + 14a3z5 - 8a3z7 + 4a4z2 - 12a4z4 + 16a4z6 - 9a4z8 - 2a5z-1 + 10a5z - 24a5z3 + 22a5z5 - 3a5z7 - 5a5z9 - 2a6 + 13a6z2 - 38a6z4 + 41a6z6 - 16a6z8 - a6z10 - 3a7z-1 + 16a7z - 31a7z3 + 20a7z5 + 5a7z7 - 8a7z9 + 6a8z2 - 21a8z4 + 28a8z6 - 11a8z8 - a8z10 - 3a9z-1 + 13a9z - 23a9z3 + 21a9z5 - 3a9z7 - 3a9z9 + 2a10 - 6a10z2 + 3a10z4 + 6a10z6 - 4a10z8 - a11z-1 + 3a11z - 7a11z3 + 8a11z5 - 3a11z7 + a12 - 3a12z2 + 3a12z4 - a12z6

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 2           1
j = 0          3 
j = -2         61 
j = -4        84  
j = -6       105   
j = -8      108    
j = -10     1010     
j = -12    711      
j = -14   59       
j = -16  27        
j = -18 15         
j = -20 2          
j = -221           


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, 62]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 62]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 13, 5, 14], X[14, 7, 15, 8], 
 
>   X[20, 18, 21, 17], X[10, 15, 11, 16], X[8, 19, 9, 20], X[18, 9, 19, 10], 
 
>   X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 8, -6, 11, -2, 3, -4, 6, -9, 5, -8, 
 
>    7, -5, 9, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(21/2)     3       7      12      16      20      20     18     13     9
q        - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 
            19/2    17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
           q       q       q       q       q       q      q      q      q
 
       4
>   ------- - Sqrt[q]
    Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -34    2     -30    2     5     4     2     3     2     -10   4    3
-2 - q    - --- + q    - --- + --- + --- - --- + --- - --- + q    + -- - -- + 
             32           26    24    18    16    14    12           8    6
            q            q     q     q     q     q     q            q    q
 
    4     2
>   -- + q
     4
    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 62]][a, z]
Out[8]=   
   3       5      7      9    11
  a     2 a    3 a    3 a    a        3        5        7        9        3
-(--) + ---- - ---- + ---- - --- - 3 a  z + 4 a  z - 7 a  z + 4 a  z - a z  - 
  z      z      z      z      z
 
     3  3      5  3      7  3    3  5      5  5
>   a  z  + 3 a  z  - 5 a  z  + a  z  + 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 62]][a, z]
Out[9]=   
                       3      5      7      9    11
    6      10    12   a    2 a    3 a    3 a    a        3         5
-2 a  + 2 a   + a   - -- - ---- - ---- - ---- - --- + 4 a  z + 10 a  z + 
                      z     z      z      z      z
 
        7         9        11        4  2       6  2      8  2      10  2
>   16 a  z + 13 a  z + 3 a   z + 4 a  z  + 13 a  z  + 6 a  z  - 6 a   z  - 
 
       12  2      3      3  3       5  3       7  3       9  3      11  3
>   3 a   z  + a z  - 8 a  z  - 24 a  z  - 31 a  z  - 23 a  z  - 7 a   z  + 
 
       2  4       4  4       6  4       8  4      10  4      12  4      5
>   5 a  z  - 12 a  z  - 38 a  z  - 21 a  z  + 3 a   z  + 3 a   z  - a z  + 
 
        3  5       5  5       7  5       9  5      11  5      2  6       4  6
>   14 a  z  + 22 a  z  + 20 a  z  + 21 a  z  + 8 a   z  - 4 a  z  + 16 a  z  + 
 
        6  6       8  6      10  6    12  6      3  7      5  7      7  7
>   41 a  z  + 28 a  z  + 6 a   z  - a   z  - 8 a  z  - 3 a  z  + 5 a  z  - 
 
       9  7      11  7      4  8       6  8       8  8      10  8      5  9
>   3 a  z  - 3 a   z  - 9 a  z  - 16 a  z  - 11 a  z  - 4 a   z  - 5 a  z  - 
 
       7  9      9  9    6  10    8  10
>   8 a  z  - 3 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
4    6      1        2        1        5        2        7        5
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    22  9    20  8    18  8    18  7    16  7    16  6    14  6
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      9        7        11       10       10      10       8      10      5
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 
     14  5    12  5    12  4    10  4    10  3    8  3    8  2    6  2    6
    q   t    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
     8           t     2  2
>   ---- + 3 t + -- + q  t
     4            2
    q  t         q


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