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L11n402

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Acknowledgement

L11n402 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X22,16,19,15 X20,8,21,7 X8,20,9,19 X13,18,14,5 X11,14,12,15 X17,12,18,13 X16,22,17,21 X2536 X4,9,1,10

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

Jones Polynomial: - 2q-6 + 5q-5 - 8q-4 + 11q-3 - 10q-2 + 11q-1 - 8 + 6q - 2q2 + q3

A2 (sl(3)) Invariant: - q-20 - 3q-18 - 3q-14 + 4q-10 + 3q-8 + 9q-6 + 4q-4 + 6q-2 + 3 + 3q4 + q8 + q10

HOMFLY-PT Polynomial: 2a-2 + a-2z2 + 2z-2 - 4z2 - 2z4 - 5a2z-2 - 7a2 - 2a2z2 + 2a2z4 + a2z6 + 4a4z-2 + 6a4 + a4z2 - a4z4 - a6z-2 - a6

Kauffman Polynomial: - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6 + a-1z + 2a-1z3 - 5a-1z5 + 2a-1z7 - 2z-2 + 7 - 9z2 + 8z4 - 8z6 + 3z8 + 5az-1 - 18az + 28az3 - 25az5 + 6az7 + az9 - 5a2z-2 + 21a2 - 34a2z2 + 27a2z4 - 19a2z6 + 7a2z8 + 9a3z-1 - 33a3z + 43a3z3 - 29a3z5 + 8a3z7 + a3z9 - 4a4z-2 + 16a4 - 24a4z2 + 19a4z4 - 9a4z6 + 4a4z8 + 5a5z-1 - 17a5z + 20a5z3 - 9a5z5 + 4a5z7 - a6z-2 + 3a6 - 4a6z2 + 4a6z4 + a6z6 + a7z-1 - 3a7z + 3a7z3

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 7         1
j = 5        21
j = 3       4  
j = 1      42  
j = -1     74   
j = -3    56    
j = -5   65     
j = -7  25      
j = -9 36       
j = -11 3        
j = -132         


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, NonAlternating, 402]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 402]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 16, 19, 15], X[20, 8, 21, 7], 
 
>   X[8, 20, 9, 19], X[13, 18, 14, 5], X[11, 14, 12, 15], X[17, 12, 18, 13], 
 
>   X[16, 22, 17, 21], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {5, -4, 9, -3}, 
 
>   {10, -1, 4, -5, 11, -2, -7, 8, -6, 7, 3, -9, -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
     2    5    8    11   10   11            2    3
-8 - -- + -- - -- + -- - -- + -- + 6 q - 2 q  + q
      6    5    4    3    2   q
     q    q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -20    3     3     4    3    9    4    6       4    8    10
3 - q    - --- - --- + --- + -- + -- + -- + -- + 3 q  + q  + q
            18    14    10    8    6    4    2
           q     q     q     q    q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 402]][a, z]
Out[8]=   
                                2      4    6           2
2       2      4    6   2    5 a    4 a    a       2   z       2  2    4  2
-- - 7 a  + 6 a  - a  + -- - ---- + ---- - -- - 4 z  + -- - 2 a  z  + a  z  - 
 2                       2     2      2     2           2
a                       z     z      z     z           a
 
       4      2  4    4  4    2  6
>   2 z  + 2 a  z  - a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 402]][a, z]
Out[9]=   
                                        2      4    6            3      5
    2        2       4      6   2    5 a    4 a    a    5 a   9 a    5 a
7 - -- + 21 a  + 16 a  + 3 a  - -- - ---- - ---- - -- + --- + ---- + ---- + 
     2                           2     2      2     2    z     z      z
    a                           z     z      z     z
 
     7                                                       2
    a    z                3         5        7        2   5 z        2  2
>   -- + - - 18 a z - 33 a  z - 17 a  z - 3 a  z - 9 z  + ---- - 34 a  z  - 
    z    a                                                  2
                                                           a
 
                            3
        4  2      6  2   2 z          3       3  3       5  3      7  3
>   24 a  z  - 4 a  z  + ---- + 28 a z  + 43 a  z  + 20 a  z  + 3 a  z  + 
                          a
 
              4                                      5
       4   4 z        2  4       4  4      6  4   5 z          5       3  5
>   8 z  - ---- + 27 a  z  + 19 a  z  + 4 a  z  - ---- - 25 a z  - 29 a  z  - 
             2                                     a
            a
 
                      6                                   7
       5  5      6   z        2  6      4  6    6  6   2 z         7
>   9 a  z  - 8 z  + -- - 19 a  z  - 9 a  z  + a  z  + ---- + 6 a z  + 
                      2                                 a
                     a
 
       3  7      5  7      8      2  8      4  8      9    3  9
>   8 a  z  + 4 a  z  + 3 z  + 7 a  z  + 4 a  z  + a z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
6    7     2        3        3       6       2       5       6      5
-- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 
 3   q    13  5    11  4    9  4    9  3    7  3    7  2    5  2    5
q        q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
     5     4 t                2      3  2      5  3    5  4    7  4
>   ---- + --- + 4 q t + 2 q t  + 4 q  t  + 2 q  t  + q  t  + q  t
     3      q
    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n402
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