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L11a201

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Acknowledgement

L11a201 as Morse Link
DrawMorseLink

PD Presentation: X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X22,18,7,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X6718 X4,15,5,16

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

Jones Polynomial: - q-15/2 + 3q-13/2 - 6q-11/2 + 8q-9/2 - 11q-7/2 + 11q-5/2 - 11q-3/2 + 9q-1/2 - 7q1/2 + 4q3/2 - 2q5/2 + q7/2

A2 (sl(3)) Invariant: q-22 - q-20 + q-18 + q-16 + q-14 + 4q-12 + 3q-8 + 2q-2 - 1 + q2 - q4 - q6 - q10

HOMFLY-PT Polynomial: a-1z-1 + 4a-1z + 4a-1z3 + a-1z5 - 2az-1 - 8az - 9az3 - 5az5 - az7 - a3z - 4a3z3 - 4a3z5 - a3z7 + a5z-1 + 2a5z + 3a5z3 + a5z5

Kauffman Polynomial: - 2a-2 + 9a-2z2 - 12a-2z4 + 6a-2z6 - a-2z8 + a-1z-1 - 4a-1z + 14a-1z3 - 20a-1z5 + 11a-1z7 - 2a-1z9 - 5 + 26z2 - 39z4 + 18z6 - z10 + 2az-1 - 10az + 21az3 - 35az5 + 26az7 - 6az9 - 3a2 + 16a2z2 - 41a2z4 + 34a2z6 - 6a2z8 - a2z10 + a3z - 9a3z3 + 9a3z5 + 6a3z7 - 4a3z9 + a4 - 3a4z2 - a4z4 + 14a4z6 - 7a4z8 - a5z-1 + 5a5z - 10a5z3 + 18a5z5 - 9a5z7 - 2a6z2 + 10a6z4 - 8a6z6 - 2a7z + 5a7z3 - 6a7z5 - 3a8z4 - a9z3

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 8           1
j = 6          1 
j = 4         31 
j = 2        41  
j = 0       53   
j = -2      64    
j = -4     66     
j = -6    55      
j = -8   36       
j = -10  35        
j = -12  3         
j = -1413          
j = -161           


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, 201]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 201]]
Out[4]=   
PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], 
 
>   X[22, 18, 7, 17], X[18, 12, 19, 11], X[12, 22, 13, 21], X[20, 14, 21, 13], 
 
>   X[14, 20, 15, 19], X[6, 7, 1, 8], X[4, 15, 5, 16]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -11, 4, -10}, 
 
>   {10, -1, 2, -3, 6, -7, 8, -9, 11, -4, 5, -6, 9, -8, 7, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(15/2)     3       6      8      11     11     11       9
-q        + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 7 Sqrt[q] + 
             13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q       q      q      q      q
 
       3/2      5/2    7/2
>   4 q    - 2 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -22    -20    -18    -16    -14    4    3    2     2    4    6    10
-1 + q    - q    + q    + q    + q    + --- + -- + -- + q  - q  - q  - q
                                         12    8    2
                                        q     q    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 201]][a, z]
Out[8]=   
             5                                    3
 1    2 a   a    4 z            3        5     4 z         3      3  3
--- - --- + -- + --- - 8 a z - a  z + 2 a  z + ---- - 9 a z  - 4 a  z  + 
a z    z    z     a                             a
 
               5
       5  3   z         5      3  5    5  5      7    3  7
>   3 a  z  + -- - 5 a z  - 4 a  z  + a  z  - a z  - a  z
              a
In[9]:=
Kauffman[Link[11, Alternating, 201]][a, z]
Out[9]=   
                                   5
     2       2    4    1    2 a   a    4 z             3        5        7
-5 - -- - 3 a  + a  + --- + --- - -- - --- - 10 a z + a  z + 5 a  z - 2 a  z + 
      2               a z    z    z     a
     a
 
               2                                      3
        2   9 z        2  2      4  2      6  2   14 z          3      3  3
>   26 z  + ---- + 16 a  z  - 3 a  z  - 2 a  z  + ----- + 21 a z  - 9 a  z  - 
              2                                     a
             a
 
                                             4
        5  3      7  3    9  3       4   12 z        2  4    4  4       6  4
>   10 a  z  + 5 a  z  - a  z  - 39 z  - ----- - 41 a  z  - a  z  + 10 a  z  - 
                                           2
                                          a
 
                  5                                                       6
       8  4   20 z          5      3  5       5  5      7  5       6   6 z
>   3 a  z  - ----- - 35 a z  + 9 a  z  + 18 a  z  - 6 a  z  + 18 z  + ---- + 
                a                                                        2
                                                                        a
 
                                        7                                  8
        2  6       4  6      6  6   11 z          7      3  7      5  7   z
>   34 a  z  + 14 a  z  - 8 a  z  + ----- + 26 a z  + 6 a  z  - 9 a  z  - -- - 
                                      a                                    2
                                                                          a
 
                           9
       2  8      4  8   2 z         9      3  9    10    2  10
>   6 a  z  - 7 a  z  - ---- - 6 a z  - 4 a  z  - z   - a  z
                         a
In[10]:=
Kh[L][q, t]
Out[10]=   
6    6      1        1        3        3        3        5        3       6
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
 4    2    16  6    14  6    14  5    12  4    10  4    10  3    8  3    8  2
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      5      5      6           4 t      2      2  2    2  3      4  3
>   ----- + ---- + ---- + 5 t + --- + 3 t  + 4 q  t  + q  t  + 3 q  t  + 
     6  2    6      4            2
    q  t    q  t   q  t         q
 
     4  4    6  4    8  5
>   q  t  + q  t  + q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a201
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