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L11n381

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Acknowledgement

L11n381 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13

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

Jones Polynomial: - q-7 + q-6 - q-5 + q-3 + q-2 + 2q-1 + q

A2 (sl(3)) Invariant: - q-22 - q-20 - 2q-18 - 2q-16 - 2q-14 - q-12 + 3q-10 + 5q-8 + 8q-6 + 7q-4 + 6q-2 + 4 + 2q2 + q4

HOMFLY-PT Polynomial: 2z-2 + 3 + z2 - 5a2z-2 - 8a2 - 5a2z2 - a2z4 + 4a4z-2 + 7a4 + 5a4z2 + a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial: - 2z-2 + 5 - 4z2 + z4 + 5az-1 - 8az + 2az3 - 5a2z-2 + 11a2 - 15a2z2 + 7a2z4 - a2z6 + 9a3z-1 - 21a3z + 20a3z3 - 8a3z5 + a3z7 - 4a4z-2 + 10a4 - 15a4z2 + 15a4z4 - 7a4z6 + a4z8 + 5a5z-1 - 19a5z + 28a5z3 - 14a5z5 + 2a5z7 - a6z-2 + 3a6 - 4a6z2 + 9a6z4 - 6a6z6 + a6z8 + a7z-1 - 6a7z + 10a7z3 - 6a7z5 + a7z7

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 3         1
j = 1       1  
j = -1      141 
j = -3     113  
j = -5    13    
j = -7   111    
j = -9   12     
j = -11 11       
j = -13          
j = -151         


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, 381]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 381]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[5, 10, 6, 11], 
 
>   X[8, 4, 9, 3], X[22, 18, 19, 17], X[11, 20, 12, 21], X[19, 12, 20, 13], 
 
>   X[18, 22, 5, 21], X[9, 16, 10, 17], X[2, 14, 3, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, 5, -3}, {-8, 7, 9, -6}, 
 
>   {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, 6, -9}]
In[6]:=
Jones[L][q]
Out[6]=   
  -7    -6    -5    -3    -2   2
-q   + q   - q   + q   + q   + - + q
                               q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -22    -20    2     2     2     -12    3    5    8    7    6       2    4
4 - q    - q    - --- - --- - --- - q    + --- + -- + -- + -- + -- + 2 q  + q
                   18    16    14           10    8    6    4    2
                  q     q     q            q     q    q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 381]][a, z]
Out[8]=   
                                 2      4    6
       2      4      6   2    5 a    4 a    a     2      2  2      4  2
3 - 8 a  + 7 a  - 2 a  + -- - ---- + ---- - -- + z  - 5 a  z  + 5 a  z  - 
                          2     2      2     2
                         z     z      z     z
 
     6  2    2  4    4  4
>   a  z  - a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 381]][a, z]
Out[9]=   
                                   2      4    6            3      5    7
        2       4      6   2    5 a    4 a    a    5 a   9 a    5 a    a
5 + 11 a  + 10 a  + 3 a  - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 
                            2     2      2     2    z     z      z     z
                           z     z      z     z
 
                3         5        7        2       2  2       4  2      6  2
>   8 a z - 21 a  z - 19 a  z - 6 a  z - 4 z  - 15 a  z  - 15 a  z  - 4 a  z  + 
 
         3       3  3       5  3       7  3    4      2  4       4  4
>   2 a z  + 20 a  z  + 28 a  z  + 10 a  z  + z  + 7 a  z  + 15 a  z  + 
 
       6  4      3  5       5  5      7  5    2  6      4  6      6  6
>   9 a  z  - 8 a  z  - 14 a  z  - 6 a  z  - a  z  - 7 a  z  - 6 a  z  + 
 
     3  7      5  7    7  7    4  8    6  8
>   a  z  + 2 a  z  + a  z  + a  z  + a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3    4         1        1        1        1       1       2       1       1
-- + - + q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 
 3   q        15  7    11  6    11  5    9  4    7  4    9  3    7  3    5  3
q            q   t    q   t    q   t    q  t    q  t    q  t    q  t    q  t
 
      1       3       1      1      1    t    3  2
>   ----- + ----- + ----- + ---- + --- + - + q  t
     7  2    5  2    3  2    3     q t   q
    q  t    q  t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n381
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