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L11n280

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Acknowledgement

L11n280 as Morse Link
DrawMorseLink

PD Presentation: X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5

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

Jones Polynomial: q-6 - 3q-5 + 6q-4 - 6q-3 + 9q-2 - 7q-1 + 8 - 5q + 2q2 - q3

A2 (sl(3)) Invariant: q-18 - q-16 + 2q-14 + 3q-12 + 4q-10 + 9q-8 + 6q-6 + 7q-4 + 3q-2 - 4q4 - q6 - q8 - q10

HOMFLY-PT Polynomial: - a-2z-2 - 2a-2 - a-2z2 + 4z-2 + 8 + 6z2 + 2z4 - 5a2z-2 - 8a2 - 7a2z2 - 4a2z4 - a2z6 + 2a4z-2 + 2a4 + 2a4z2 + a4z4

Kauffman Polynomial: a-3z-1 - 2a-3z + a-3z3 - a-2z-2 + a-2 - 2a-2z2 + 2a-2z4 + 5a-1z-1 - 11a-1z + 11a-1z3 - 3a-1z5 + a-1z7 - 4z-2 + 7 - 9z2 + 14z4 - 7z6 + 2z8 + 9az-1 - 21az + 23az3 - 11az5 + az7 + az9 - 5a2z-2 + 10a2 - 11a2z2 + 15a2z4 - 15a2z6 + 5a2z8 + 5a3z-1 - 12a3z + 18a3z3 - 17a3z5 + 3a3z7 + a3z9 - 2a4z-2 + 4a4 - a4z2 - 7a4z6 + 3a4z8 + 5a5z3 - 9a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 7         1
j = 5        1 
j = 3       41 
j = 1      41  
j = -1     56   
j = -3    431   
j = -5   25     
j = -7  44      
j = -9 14       
j = -11 2        
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, 280]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 280]]
Out[4]=   
PD[X[6, 1, 7, 2], X[5, 12, 6, 13], X[8, 4, 9, 3], X[2, 14, 3, 13], 
 
>   X[14, 7, 15, 8], X[9, 18, 10, 19], X[17, 11, 18, 22], X[11, 21, 12, 20], 
 
>   X[21, 17, 22, 16], X[4, 15, 1, 16], X[19, 10, 20, 5]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, 
 
>   {-8, 2, 4, -5, 10, 9, -7, 6, -11, 8, -9, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
     -6   3    6    6    9    7            2    3
8 + q   - -- + -- - -- + -- - - - 5 q + 2 q  - q
           5    4    3    2   q
          q    q    q    q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 -18    -16    2     3     4    9    6    7    3       4    6    8    10
q    - q    + --- + --- + --- + -- + -- + -- + -- - 4 q  - q  - q  - q
               14    12    10    8    6    4    2
              q     q     q     q    q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 280]][a, z]
Out[8]=   
                                       2      4           2
    2       2      4   4      1     5 a    2 a       2   z       2  2
8 - -- - 8 a  + 2 a  + -- - ----- - ---- + ---- + 6 z  - -- - 7 a  z  + 
     2                  2    2  2     2      2            2
    a                  z    a  z     z      z            a
 
       4  2      4      2  4    4  4    2  6
>   2 a  z  + 2 z  - 4 a  z  + a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 280]][a, z]
Out[9]=   
                                              2      4
     -2       2      4    6   4      1     5 a    2 a     1      5    9 a
7 + a   + 10 a  + 4 a  - a  - -- - ----- - ---- - ---- + ---- + --- + --- + 
                               2    2  2     2      2     3     a z    z
                              z    a  z     z      z     a  z
 
       3                                             2
    5 a    2 z   11 z                3        2   2 z        2  2    4  2
>   ---- - --- - ---- - 21 a z - 12 a  z - 9 z  - ---- - 11 a  z  - a  z  + 
     z      3     a                                 2
           a                                       a
 
               3       3                                             4
       6  2   z    11 z          3       3  3      5  3       4   2 z
>   3 a  z  + -- + ----- + 23 a z  + 18 a  z  + 5 a  z  + 14 z  + ---- + 
               3     a                                              2
              a                                                    a
 
                            5
        2  4      6  4   3 z          5       3  5      5  5      6
>   15 a  z  - 3 a  z  - ---- - 11 a z  - 17 a  z  - 9 a  z  - 7 z  - 
                          a
 
                                  7
        2  6      4  6    6  6   z       7      3  7      5  7      8
>   15 a  z  - 7 a  z  + a  z  + -- + a z  + 3 a  z  + 3 a  z  + 2 z  + 
                                 a
 
       2  8      4  8      9    3  9
>   5 a  z  + 3 a  z  + a z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -3   6           1        2        1       4       4       4       2
q   + - + 4 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 
      q          13  6    11  5    9  5    9  4    7  4    7  3    5  3
                q   t    q   t    q  t    q  t    q  t    q  t    q  t
 
      5       4      3      5             3      3  2    5  2    7  3
>   ----- + ----- + ---- + --- + q t + 4 q  t + q  t  + q  t  + q  t
     5  2    3  2    3     q t
    q  t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n280
L11n279
L11n279
L11n281
L11n281