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L11a173

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Acknowledgement

L11a173 as Morse Link
DrawMorseLink

PD Presentation: X8192 X2,9,3,10 X10,3,11,4 X20,14,21,13 X14,6,15,5 X4,21,5,22 X16,7,17,8 X22,15,7,16 X18,12,19,11 X12,20,13,19 X6,18,1,17

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

Jones Polynomial: - q-13/2 + 3q-11/2 - 6q-9/2 + 10q-7/2 - 14q-5/2 + 15q-3/2 - 16q-1/2 + 13q1/2 - 10q3/2 + 6q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant: q-18 - q-16 + 2q-14 - 2q-12 + q-10 + 2q-8 - q-6 + 5q-4 - 2q-2 + 4 + 2q6 - 2q8 + q10 - q12

HOMFLY-PT Polynomial: 4a-1z + 8a-1z3 + 5a-1z5 + a-1z7 - az-1 - 9az - 20az3 - 18az5 - 7az7 - az9 + a3z-1 + 4a3z + 8a3z3 + 5a3z5 + a3z7

Kauffman Polynomial: - 2a-4z2 + 3a-4z4 - a-4z6 - 6a-3z3 + 9a-3z5 - 3a-3z7 + a-2z2 - 5a-2z4 + 10a-2z6 - 4a-2z8 - 6a-1z + 17a-1z3 - 15a-1z5 + 11a-1z7 - 4a-1z9 + 6z2 - 8z4 + 5z6 + z8 - 2z10 + az-1 - 11az + 39az3 - 53az5 + 32az7 - 9az9 - a2 + 8a2z2 - 21a2z4 + 12a2z6 - a2z8 - 2a2z10 + a3z-1 - 3a3z + 7a3z3 - 17a3z5 + 13a3z7 - 5a3z9 + 4a4z2 - 15a4z4 + 15a4z6 - 6a4z8 + 2a5z - 7a5z3 + 11a5z5 - 5a5z7 - a6z2 + 6a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 10           1
j = 8          2 
j = 6         41 
j = 4        62  
j = 2       74   
j = 0      96    
j = -2     78     
j = -4    78      
j = -6   48       
j = -8  26        
j = -10 14         
j = -12 2          
j = -141           


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, 173]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 173]]
Out[4]=   
PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[20, 14, 21, 13], 
 
>   X[14, 6, 15, 5], X[4, 21, 5, 22], X[16, 7, 17, 8], X[22, 15, 7, 16], 
 
>   X[18, 12, 19, 11], X[12, 20, 13, 19], X[6, 18, 1, 17]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -6, 5, -11}, 
 
>   {7, -1, 2, -3, 9, -10, 4, -5, 8, -7, 11, -9, 10, -4, 6, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(13/2)     3      6      10     14     15      16
-q        + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 
             11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q      q      q      q
 
        3/2      5/2      7/2    9/2
>   10 q    + 6 q    - 3 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -18    -16    2     2     -10   2     -6   5    2       6      8    10
4 + q    - q    + --- - --- + q    + -- - q   + -- - -- + 2 q  - 2 q  + q   - 
                   14    12           8          4    2
                  q     q            q          q    q
 
     12
>   q
In[8]:=
HOMFLYPT[Link[11, Alternating, 173]][a, z]
Out[8]=   
        3                             3                          5
  a    a    4 z              3     8 z          3      3  3   5 z          5
-(-) + -- + --- - 9 a z + 4 a  z + ---- - 20 a z  + 8 a  z  + ---- - 18 a z  + 
  z    z     a                      a                          a
 
               7
       3  5   z         7    3  7      9
>   5 a  z  + -- - 7 a z  + a  z  - a z
              a
In[9]:=
Kauffman[Link[11, Alternating, 173]][a, z]
Out[9]=   
           3                                              2    2
  2   a   a    6 z               3        5        2   2 z    z       2  2
-a  + - + -- - --- - 11 a z - 3 a  z + 2 a  z + 6 z  - ---- + -- + 8 a  z  + 
      z   z     a                                        4     2
                                                        a     a
 
                         3       3
       4  2    6  2   6 z    17 z          3      3  3      5  3      7  3
>   4 a  z  - a  z  - ---- + ----- + 39 a z  + 7 a  z  - 7 a  z  + 2 a  z  - 
                        3      a
                       a
 
              4      4                                      5       5
       4   3 z    5 z        2  4       4  4      6  4   9 z    15 z
>   8 z  + ---- - ---- - 21 a  z  - 15 a  z  + 6 a  z  + ---- - ----- - 
             4      2                                      3      a
            a      a                                      a
 
                                                    6       6
          5       3  5       5  5    7  5      6   z    10 z        2  6
>   53 a z  - 17 a  z  + 11 a  z  - a  z  + 5 z  - -- + ----- + 12 a  z  + 
                                                    4     2
                                                   a     a
 
                            7       7
        4  6      6  6   3 z    11 z          7       3  7      5  7    8
>   15 a  z  - 3 a  z  - ---- + ----- + 32 a z  + 13 a  z  - 5 a  z  + z  - 
                           3      a
                          a
 
       8                        9
    4 z     2  8      4  8   4 z         9      3  9      10      2  10
>   ---- - a  z  - 6 a  z  - ---- - 9 a z  - 5 a  z  - 2 z   - 2 a  z
      2                       a
     a
In[10]:=
Kh[L][q, t]
Out[10]=   
    8      1        2        1        4        2       6       4       8
9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
     2    14  6    12  5    10  5    10  4    8  4    8  3    6  3    6  2
    q    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      7      8      7              2        2  2      4  2      4  3
>   ----- + ---- + ---- + 6 t + 7 q  t + 4 q  t  + 6 q  t  + 2 q  t  + 
     4  2    4      2
    q  t    q  t   q  t
 
       6  3    6  4      8  4    10  5
>   4 q  t  + q  t  + 2 q  t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a173
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