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L11n321

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Acknowledgement

L11n321 as Morse Link
DrawMorseLink

PD Presentation: X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20

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

Jones Polynomial: q-6 - 2q-5 + 4q-4 - 6q-3 + 9q-2 - 8q-1 + 9 - 6q + 5q2 - 2q3

A2 (sl(3)) Invariant: q-18 + 2q-14 + 2q-12 + 4q-8 + 2q-6 + 4q-4 + 5q-2 + 3 + 5q2 + q6 - 2q10

HOMFLY-PT Polynomial: - 2a-2 - 2a-2z2 + z-2 + 8 + 9z2 + 3z4 - 2a2z-2 - 9a2 - 11a2z2 - 5a2z4 - a2z6 + a4z-2 + 3a4 + 3a4z2 + a4z4

Kauffman Polynomial: - 2a-3z + 3a-3z3 + 3a-2 - 7a-2z2 + 4a-2z4 + a-2z6 - 7a-1z + 11a-1z3 - 6a-1z5 + 3a-1z7 - z-2 + 11 - 24z2 + 22z4 - 9z6 + 3z8 + 2az-1 - 12az + 15az3 - 9az5 + 2az7 + az9 - 2a2z-2 + 11a2 - 23a2z2 + 24a2z4 - 16a2z6 + 5a2z8 + 2a3z-1 - 8a3z + 13a3z3 - 10a3z5 + a3z7 + a3z9 - a4z-2 + 3a4 - 2a4z2 + 2a4z4 - 5a4z6 + 2a4z8 - a5z + 6a5z3 - 7a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + a6z6

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 7         2
j = 5        3 
j = 3       32 
j = 1      63  
j = -1     56   
j = -3    43    
j = -5   25     
j = -7  24      
j = -9 13       
j = -11 1        
j = -131         


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, 321]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 321]]
Out[4]=   
PD[X[6, 1, 7, 2], X[5, 14, 6, 15], X[3, 8, 4, 9], X[2, 16, 3, 15], 
 
>   X[16, 7, 17, 8], X[9, 18, 10, 19], X[17, 1, 18, 4], X[19, 13, 20, 22], 
 
>   X[13, 10, 14, 11], X[21, 5, 22, 12], X[11, 21, 12, 20]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, -3, 7}, {-2, -1, 5, 3, -6, 9, -11, 10}, 
 
>   {-9, 2, 4, -5, -7, 6, -8, 11, -10, 8}]
In[6]:=
Jones[L][q]
Out[6]=   
     -6   2    4    6    9    8            2      3
9 + q   - -- + -- - -- + -- - - - 6 q + 5 q  - 2 q
           5    4    3    2   q
          q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -18    2     2    4    2    4    5       2    6      10
3 + q    + --- + --- + -- + -- + -- + -- + 5 q  + q  - 2 q
            14    12    8    6    4    2
           q     q     q    q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 321]][a, z]
Out[8]=   
                                2    4             2
    2       2      4    -2   2 a    a       2   2 z        2  2      4  2
8 - -- - 9 a  + 3 a  + z   - ---- + -- + 9 z  - ---- - 11 a  z  + 3 a  z  + 
     2                         2     2            2
    a                         z     z            a
 
       4      2  4    4  4    2  6
>   3 z  - 5 a  z  + a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 321]][a, z]
Out[9]=   
                                       2    4            3
     3        2      4    6    -2   2 a    a    2 a   2 a    2 z   7 z
11 + -- + 11 a  + 3 a  - a  - z   - ---- - -- + --- + ---- - --- - --- - 
      2                               2     2    z     z      3     a
     a                               z     z                 a
 
                                        2
                3      5         2   7 z        2  2      4  2      6  2
>   12 a z - 8 a  z - a  z - 24 z  - ---- - 23 a  z  - 2 a  z  + 4 a  z  + 
                                       2
                                      a
 
       3       3                                             4
    3 z    11 z          3       3  3      5  3       4   4 z        2  4
>   ---- + ----- + 15 a z  + 13 a  z  + 6 a  z  + 22 z  + ---- + 24 a  z  + 
      3      a                                              2
     a                                                     a
 
                           5                                         6
       4  4      6  4   6 z         5       3  5      5  5      6   z
>   2 a  z  - 4 a  z  - ---- - 9 a z  - 10 a  z  - 7 a  z  - 9 z  + -- - 
                         a                                           2
                                                                    a
 
                                    7
        2  6      4  6    6  6   3 z         7    3  7      5  7      8
>   16 a  z  - 5 a  z  + a  z  + ---- + 2 a z  + a  z  + 2 a  z  + 3 z  + 
                                  a
 
       2  8      4  8      9    3  9
>   5 a  z  + 2 a  z  + a z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
6           1        1        1       3       2       4       2       5
- + 6 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q          13  6    11  5    9  5    9  4    7  4    7  3    5  3    5  2
          q   t    q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
      4      3      5               3        3  2      5  2      7  3
>   ----- + ---- + --- + 3 q t + 3 q  t + 2 q  t  + 3 q  t  + 2 q  t
     3  2    3     q t
    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n321
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