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L11n342

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Acknowledgement

L11n342 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X11,19,12,18 X7,14,8,15 X13,8,14,9 X15,21,16,20 X17,13,18,22 X21,17,22,16 X19,5,20,12 X2536 X4,9,1,10

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

Jones Polynomial: q-5 - q-4 + 4q-3 - 3q-2 + 4q-1 - 2 + 2q - q2 + q4 - q5

A2 (sl(3)) Invariant: q-16 + 2q-14 + 3q-12 + 6q-10 + 6q-8 + 7q-6 + 4q-4 + 2q-2 + 1 - 2q2 - q6 - q12 - q16

HOMFLY-PT Polynomial: - a-4 - a-4z2 - a-2z-2 + 4z-2 + 8 + 8z2 + 5z4 + z6 - 5a2z-2 - 10a2 - 8a2z2 - 2a2z4 + 2a4z-2 + 3a4 + a4z2

Kauffman Polynomial: 2a-5z - 4a-5z3 + a-5z5 - a-4 + 4a-4z2 - 5a-4z4 + a-4z6 + a-3z-1 - 2a-3z + a-3z3 - a-3z5 - a-2z-2 + 2a-2 + a-2z2 - 3a-2z4 + 5a-1z-1 - 23a-1z + 40a-1z3 - 25a-1z5 + 4a-1z7 - 4z-2 + 17 - 33z2 + 42z4 - 24z6 + 4z8 + 9az-1 - 37az + 52az3 - 24az5 + az9 - 5a2z-2 + 24a2 - 51a2z2 + 58a2z4 - 30a2z6 + 5a2z8 + 5a3z-1 - 18a3z + 17a3z3 - a3z5 - 4a3z7 + a3z9 - 2a4z-2 + 11a4 - 21a4z2 + 18a4z4 - 7a4z6 + a4z8

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 11           1
j = 9            
j = 7        111 
j = 5       21   
j = 3      221   
j = 1     341    
j = -1    123     
j = -3   23       
j = -5  21        
j = -7 14         
j = -9            
j = -111           


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, 342]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 342]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 19, 12, 18], X[7, 14, 8, 15], 
 
>   X[13, 8, 14, 9], X[15, 21, 16, 20], X[17, 13, 18, 22], X[21, 17, 22, 16], 
 
>   X[19, 5, 20, 12], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, 
 
>   {-5, 4, -6, 8, -7, 3, -9, 6, -8, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
      -5    -4   4    3    4          2    4    5
-2 + q   - q   + -- - -- + - + 2 q - q  + q  - q
                  3    2   q
                 q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -16    2     3     6    6    7    4    2       2    6    12    16
1 + q    + --- + --- + --- + -- + -- + -- + -- - 2 q  - q  - q   - q
            14    12    10    8    6    4    2
           q     q     q     q    q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 342]][a, z]
Out[8]=   
                                         2      4           2
     -4       2      4   4      1     5 a    2 a       2   z       2  2
8 - a   - 10 a  + 3 a  + -- - ----- - ---- + ---- + 8 z  - -- - 8 a  z  + 
                          2    2  2     2      2            4
                         z    a  z     z      z            a
 
     4  2      4      2  4    6
>   a  z  + 5 z  - 2 a  z  + z
In[9]:=
Kauffman[Link[11, NonAlternating, 342]][a, z]
Out[9]=   
                                                2      4
      -4   2        2       4   4      1     5 a    2 a     1      5    9 a
17 - a   + -- + 24 a  + 11 a  - -- - ----- - ---- - ---- + ---- + --- + --- + 
            2                    2    2  2     2      2     3     a z    z
           a                    z    a  z     z      z     a  z
 
       3                                                    2    2
    5 a    2 z   2 z   23 z                3         2   4 z    z        2  2
>   ---- + --- - --- - ---- - 37 a z - 18 a  z - 33 z  + ---- + -- - 51 a  z  - 
     z      5     3     a                                  4     2
           a     a                                        a     a
 
                  3    3       3                                   4      4
        4  2   4 z    z    40 z          3       3  3       4   5 z    3 z
>   21 a  z  - ---- + -- + ----- + 52 a z  + 17 a  z  + 42 z  - ---- - ---- + 
                 5     3     a                                    4      2
                a     a                                          a      a
 
                           5    5       5                              6
        2  4       4  4   z    z    25 z          5    3  5       6   z
>   58 a  z  + 18 a  z  + -- - -- - ----- - 24 a z  - a  z  - 24 z  + -- - 
                           5    3     a                                4
                          a    a                                      a
 
                            7
        2  6      4  6   4 z       3  7      8      2  8    4  8      9    3  9
>   30 a  z  - 7 a  z  + ---- - 4 a  z  + 4 z  + 5 a  z  + a  z  + a z  + a  z
                          a
In[10]:=
Kh[L][q, t]
Out[10]=   
3            3     1        1       4       2       1       2       3
- + 4 q + 2 q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 
q                 11  6    7  5    7  4    5  4    5  3    3  3    3  2
                 q   t    q  t    q  t    q  t    q  t    q  t    q  t
 
     1      2    3 q            3        5      3  2    5  2    7  2    7  3
>   ---- + --- + --- + q t + 2 q  t + 2 q  t + q  t  + q  t  + q  t  + q  t  + 
       2   q t    t
    q t
 
     7  4    11  5
>   q  t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n342
L11n341
L11n341
L11n343
L11n343