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L11a57

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Acknowledgement

L11a57 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-11/2 + 3q-9/2 - 7q-7/2 + 12q-5/2 - 16q-3/2 + 18q-1/2 - 20q1/2 + 17q3/2 - 13q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant: q-18 + q-16 - q-14 + 2q-12 - 3q-8 + 2q-6 - 3q-4 + 2q-2 + 3 + 5q4 - 3q6 + 3q8 + q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial: a-3z + 2a-3z3 + a-3z5 - 2a-1z-1 - 5a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + 4az-1 + 9az + 9az3 + 3az5 - 3a3z-1 - 6a3z - 3a3z3 + a5z-1 + a5z

Kauffman Polynomial: - a-6z4 + 2a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 8a-4z6 + 3a-3z - 8a-3z3 + 14a-3z5 - 11a-3z7 - a-2 - 2a-2z2 + 3a-2z4 + 10a-2z6 - 10a-2z8 - 2a-1z-1 + 15a-1z - 44a-1z3 + 47a-1z5 - 10a-1z7 - 5a-1z9 - 2 + 12z2 - 38z4 + 51z6 - 18z8 - z10 - 4az-1 + 26az - 60az3 + 42az5 + 5az7 - 8az9 - 3a2 + 19a2z2 - 47a2z4 + 44a2z6 - 11a2z8 - a2z10 - 3a3z-1 + 18a3z - 32a3z3 + 17a3z5 + 3a3z7 - 3a3z9 - a4 + 7a4z2 - 14a4z4 + 11a4z6 - 3a4z8 - a5z-1 + 4a5z - 6a5z3 + 4a5z5 - a5z7

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 12           1
j = 10          3 
j = 8         51 
j = 6        83  
j = 4       95   
j = 2      118    
j = 0     911     
j = -2    79      
j = -4   59       
j = -6  27        
j = -8 15         
j = -10 2          
j = -121           


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, 57]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 57]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[18, 16, 19, 15], 
 
>   X[16, 9, 17, 10], X[8, 17, 9, 18], X[22, 20, 5, 19], X[20, 11, 21, 12], 
 
>   X[10, 21, 11, 22], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 8, -2, 11, -3, 4, -5, 6, -4, 
 
>    7, -8, 9, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(11/2)    3      7      12     16      18                       3/2
-q        + ---- - ---- + ---- - ---- + ------- - 20 Sqrt[q] + 17 q    - 
             9/2    7/2    5/2    3/2   Sqrt[q]
            q      q      q      q
 
        5/2      7/2      9/2    11/2
>   13 q    + 8 q    - 4 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -18    -16    -14    2    3    2    3    2       4      6      8    10
3 + q    + q    - q    + --- - -- + -- - -- + -- + 5 q  - 3 q  + 3 q  + q   - 
                          12    8    6    4    2
                         q     q    q    q    q
 
       12      14    16
>   2 q   + 2 q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 57]][a, z]
Out[8]=   
               3    5                                         3      3
-2    4 a   3 a    a    z    5 z              3      5     2 z    7 z
--- + --- - ---- + -- + -- - --- + 9 a z - 6 a  z + a  z + ---- - ---- + 
a z    z     z     z     3    a                              3     a
                        a                                   a
 
                        5      5             7
         3      3  3   z    4 z         5   z
>   9 a z  - 3 a  z  + -- - ---- + 3 a z  - --
                        3    a              a
                       a
In[9]:=
Kauffman[Link[11, Alternating, 57]][a, z]
Out[9]=   
                                      3    5
      -2      2    4    2    4 a   3 a    a    3 z   15 z                3
-2 - a   - 3 a  - a  - --- - --- - ---- - -- + --- + ---- + 26 a z + 18 a  z + 
                       a z    z     z     z     3     a
                                               a
 
                        2      2                           3      3       3
       5         2   2 z    2 z        2  2      4  2   2 z    8 z    44 z
>   4 a  z + 12 z  - ---- - ---- + 19 a  z  + 7 a  z  + ---- - ---- - ----- - 
                       4      2                           5      3      a
                      a      a                           a      a
 
                                            4      4      4
          3       3  3      5  3       4   z    7 z    3 z        2  4
>   60 a z  - 32 a  z  - 6 a  z  - 38 z  - -- + ---- + ---- - 47 a  z  - 
                                            6     4      2
                                           a     a      a
 
                  5       5       5
        4  4   4 z    14 z    47 z          5       3  5      5  5       6
>   14 a  z  - ---- + ----- + ----- + 42 a z  + 17 a  z  + 4 a  z  + 51 z  - 
                 5      3       a
                a      a
 
       6       6                             7       7
    8 z    10 z        2  6       4  6   11 z    10 z         7      3  7
>   ---- + ----- + 44 a  z  + 11 a  z  - ----- - ----- + 5 a z  + 3 a  z  - 
      4      2                             3       a
     a      a                             a
 
                        8                           9
     5  7       8   10 z        2  8      4  8   5 z         9      3  9
>   a  z  - 18 z  - ----- - 11 a  z  - 3 a  z  - ---- - 8 a z  - 3 a  z  - 
                      2                           a
                     a
 
     10    2  10
>   z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
         2     1        2        1       5       2       7       5       9
11 + 11 q  + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
              12  6    10  5    8  5    8  4    6  4    6  3    4  3    4  2
             q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
      7     9    9        2        4        4  2      6  2      6  3
>   ----- + - + ---- + 8 q  t + 9 q  t + 5 q  t  + 8 q  t  + 3 q  t  + 
     2  2   t    2
    q  t        q  t
 
       8  3    8  4      10  4    12  5
>   5 q  t  + q  t  + 3 q   t  + q   t


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