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L11n267

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Acknowledgement

L11n267 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X4,9,1,10

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

Jones Polynomial: q-5 - q-4 + 4q-3 - 2q-2 + 4q-1 - 1 + q - q2 - q3 + q4 - q5

A2 (sl(3)) Invariant: q-16 + 3q-14 + 4q-12 + 7q-10 + 8q-8 + 9q-6 + 6q-4 + 4q-2 + 2 - 3q2 - 3q4 - 5q6 - 3q8 - 2q10 - q12 + q14 - q16

HOMFLY-PT Polynomial: - a-4z2 - 2a-2z-2 - 7a-2 - 6a-2z2 - a-2z4 + 7z-2 + 21 + 23z2 + 12z4 + 2z6 - 8a2z-2 - 18a2 - 14a2z2 - 3a2z4 + 3a4z-2 + 4a4 + a4z2

Kauffman Polynomial: - a-5z-1 + 3a-5z - 4a-5z3 + a-5z5 + a-4 + a-4z2 - 4a-4z4 + a-4z6 - a-3z-1 + 3a-3z - 3a-3z3 - 2a-2z-2 + 7a-2 - 9a-2z2 + 3a-2z4 - a-2z6 + 7a-1z-1 - 21a-1z + 33a-1z3 - 20a-1z5 + 3a-1z7 - 7z-2 + 22 - 42z2 + 50z4 - 26z6 + 4z8 + 15az-1 - 45az + 53az3 - 21az5 - az7 + az9 - 8a2z-2 + 28a2 - 54a2z2 + 61a2z4 - 31a2z6 + 5a2z8 + 8a3z-1 - 24a3z + 21a3z3 - 2a3z5 - 4a3z7 + a3z9 - 3a4z-2 + 13a4 - 22a4z2 + 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         11 
j = 5       31   
j = 3      121   
j = 1     341    
j = -1    113     
j = -3   13       
j = -5  31        
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, 267]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 267]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], 
 
>   X[11, 19, 12, 18], X[15, 21, 16, 20], X[17, 9, 18, 22], X[21, 17, 22, 16], 
 
>   X[19, 13, 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, 3, -4}, 
 
>   {11, -2, -5, 9, 4, -3, -6, 8, -7, 5, -9, 6, -8, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
      -5    -4   4    2    4        2    3    4    5
-1 + q   - q   + -- - -- + - + q - q  - q  + q  - q
                  3    2   q
                 q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -16    3     4     7    8    9    6    4       2      4      6      8
2 + q    + --- + --- + --- + -- + -- + -- + -- - 3 q  - 3 q  - 5 q  - 3 q  - 
            14    12    10    8    6    4    2
           q     q     q     q    q    q    q
 
       10    12    14    16
>   2 q   - q   + q   - q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 267]][a, z]
Out[8]=   
                                         2      4            2      2
     7        2      4   7      2     8 a    3 a        2   z    6 z
21 - -- - 18 a  + 4 a  + -- - ----- - ---- + ---- + 23 z  - -- - ---- - 
      2                   2    2  2     2      2             4     2
     a                   z    a  z     z      z             a     a
 
                                4
        2  2    4  2       4   z       2  4      6
>   14 a  z  + a  z  + 12 z  - -- - 3 a  z  + 2 z
                                2
                               a
In[9]:=
Kauffman[Link[11, NonAlternating, 267]][a, z]
Out[9]=   
                                                2      4
      -4   7        2       4   7      2     8 a    3 a     1      1      7
22 + a   + -- + 28 a  + 13 a  - -- - ----- - ---- - ---- - ---- - ---- + --- + 
            2                    2    2  2     2      2     5      3     a z
           a                    z    a  z     z      z     a  z   a  z
 
              3                                                  2      2
    15 a   8 a    3 z   3 z   21 z                3         2   z    9 z
>   ---- + ---- + --- + --- - ---- - 45 a z - 24 a  z - 42 z  + -- - ---- - 
     z      z      5     3     a                                 4     2
                  a     a                                       a     a
 
                             3      3       3
        2  2       4  2   4 z    3 z    33 z          3       3  3       4
>   54 a  z  - 22 a  z  - ---- - ---- + ----- + 53 a z  + 21 a  z  + 50 z  - 
                            5      3      a
                           a      a
 
       4      4                          5       5
    4 z    3 z        2  4       4  4   z    20 z          5      3  5
>   ---- + ---- + 61 a  z  + 18 a  z  + -- - ----- - 21 a z  - 2 a  z  - 
      4      2                           5     a
     a      a                           a
 
             6    6                           7
        6   z    z        2  6      4  6   3 z       7      3  7      8
>   26 z  + -- - -- - 31 a  z  - 7 a  z  + ---- - a z  - 4 a  z  + 4 z  + 
             4    2                         a
            a    a
 
       2  8    4  8      9    3  9
>   5 a  z  + a  z  + a z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3          3     1        1       4       3       1       1       3      1
- + 4 q + q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 
q               11  6    7  5    7  4    5  4    5  3    3  3    3  2      2
               q   t    q  t    q  t    q  t    q  t    q  t    q  t    q t
 
     1    3 q            3        5      3  2    5  2    7  3    7  4    11  5
>   --- + --- + q t + 2 q  t + 3 q  t + q  t  + q  t  + q  t  + q  t  + q   t
    q t    t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n267
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