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L11n169

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Acknowledgement

L11n169 as Morse Link
DrawMorseLink

PD Presentation: X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,7 X21,16,22,17 X6718 X4,13,5,14

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

Jones Polynomial: q-29/2 - 2q-27/2 + 3q-25/2 - 4q-23/2 + 4q-21/2 - 3q-19/2 + 2q-17/2 - q-15/2 - q-13/2 - q-9/2

A2 (sl(3)) Invariant: - 2q-44 - q-40 + q-36 - q-34 + q-32 - q-30 + q-28 + 2q-26 + 2q-24 + 3q-22 + 2q-20 + q-18 + q-16

HOMFLY-PT Polynomial: - 3a9z-1 - 20a9z - 37a9z3 - 28a9z5 - 9a9z7 - a9z9 + 5a11z-1 + 22a11z + 24a11z3 + 9a11z5 + a11z7 - 2a13z-1 - 5a13z - 2a13z3

Kauffman Polynomial: 3a9z-1 - 20a9z + 37a9z3 - 28a9z5 + 9a9z7 - a9z9 - 5a10 + 22a10z2 - 24a10z4 + 9a10z6 - a10z8 + 5a11z-1 - 26a11z + 45a11z3 - 33a11z5 + 10a11z7 - a11z9 - 5a12 + 24a12z2 - 26a12z4 + 9a12z6 - a12z8 + 2a13z-1 - 5a13z + 3a13z3 - 2a13z5 - 3a14z2 + 3a14z4 - 2a14z6 - a15z3 + a15z5 - a15z7 + a16 - 3a16z2 + 4a16z4 - 2a16z6 - a17z + 4a17z3 - 2a17z5 + 2a18z2 - a18z4

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -8           1
j = -10           1
j = -12         1  
j = -14       2    
j = -16      111   
j = -18     32     
j = -20    221     
j = -22   22       
j = -24  12        
j = -26 12         
j = -28 1          
j = -301           


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, NonAlternating, 169]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 169]]
Out[4]=   
PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[14, 5, 15, 6], 
 
>   X[15, 20, 16, 21], X[11, 18, 12, 19], X[19, 12, 20, 13], X[17, 22, 18, 7], 
 
>   X[21, 16, 22, 17], X[6, 7, 1, 8], X[4, 13, 5, 14]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -11, 4, -10}, 
 
>   {10, -1, 2, -3, -6, 7, 11, -4, -5, 9, -8, 6, -7, 5, -9, 8}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(29/2)     2       3       4       4       3       2      -(15/2)
q        - ----- + ----- - ----- + ----- - ----- + ----- - q        - 
            27/2    25/2    23/2    21/2    19/2    17/2
           q       q       q       q       q       q
 
     -(13/2)    -(9/2)
>   q        - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
-2     -40    -36    -34    -32    -30    -28    2     2     3     2     -18
--- - q    + q    - q    + q    - q    + q    + --- + --- + --- + --- + q    + 
 44                                              26    24    22    20
q                                               q     q     q     q
 
     -16
>   q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 169]][a, z]
Out[8]=   
    9      11      13
-3 a    5 a     2 a         9         11        13         9  3       11  3
----- + ----- - ----- - 20 a  z + 22 a   z - 5 a   z - 37 a  z  + 24 a   z  - 
  z       z       z
 
       13  3       9  5      11  5      9  7    11  7    9  9
>   2 a   z  - 28 a  z  + 9 a   z  - 9 a  z  + a   z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 169]][a, z]
Out[9]=   
                          9      11      13
    10      12    16   3 a    5 a     2 a         9         11        13
-5 a   - 5 a   + a   + ---- + ----- + ----- - 20 a  z - 26 a   z - 5 a   z - 
                        z       z       z
 
     17         10  2       12  2      14  2      16  2      18  2       9  3
>   a   z + 22 a   z  + 24 a   z  - 3 a   z  - 3 a   z  + 2 a   z  + 37 a  z  + 
 
        11  3      13  3    15  3      17  3       10  4       12  4
>   45 a   z  + 3 a   z  - a   z  + 4 a   z  - 24 a   z  - 26 a   z  + 
 
       14  4      16  4    18  4       9  5       11  5      13  5    15  5
>   3 a   z  + 4 a   z  - a   z  - 28 a  z  - 33 a   z  - 2 a   z  + a   z  - 
 
       17  5      10  6      12  6      14  6      16  6      9  7
>   2 a   z  + 9 a   z  + 9 a   z  - 2 a   z  - 2 a   z  + 9 a  z  + 
 
        11  7    15  7    10  8    12  8    9  9    11  9
>   10 a   z  - a   z  - a   z  - a   z  - a  z  - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -10    -8      1         1         1        2        1        2        2
q    + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
              30  11    28  10    26  10    26  9    24  9    24  8    22  8
             q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      2        2        2        3        1        2        1        1
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     22  7    20  7    20  6    18  6    20  5    18  5    16  5    16  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      2        1        1
>   ------ + ------ + ------
     14  4    16  3    12  2
    q   t    q   t    q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n169
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