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L11n223

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Acknowledgement

L11n223 as Morse Link
DrawMorseLink

PD Presentation: X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,8,19,7 X16,9,17,10 X22,15,9,16 X21,5,22,4 X5,14,6,15 X13,20,14,21 X8,18,1,17 X6,20,7,19

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

Jones Polynomial: q-17/2 - 2q-15/2 + 4q-13/2 - 7q-11/2 + 8q-9/2 - 8q-7/2 + 7q-5/2 - 6q-3/2 + 3q-1/2 - 2q1/2

A2 (sl(3)) Invariant: - q-26 - q-20 + 2q-18 - q-16 - q-10 + 3q-8 + 3q-4 + 2q-2 + 1 + 2q2

HOMFLY-PT Polynomial: - 2az-1 - 5az - 2az3 + 3a3z-1 + 7a3z + 7a3z3 + 2a3z5 - a5z-1 + 2a5z3 + a5z5 - 2a7z - a7z3

Kauffman Polynomial: 2az-1 - 9az + 10az3 - 3az5 - 3a2 + 6a2z2 - 2a2z4 + 2a2z6 - a2z8 + 3a3z-1 - 13a3z + 21a3z3 - 10a3z5 + 3a3z7 - a3z9 - 3a4 + 16a4z2 - 20a4z4 + 13a4z6 - 4a4z8 + a5z-1 - 7a5z3 + 6a5z5 - a5z7 - a5z9 - a6 + 5a6z2 - 12a6z4 + 8a6z6 - 3a6z8 + 4a7z - 15a7z3 + 11a7z5 - 4a7z7 - 3a8z2 + 5a8z4 - 3a8z6 + 3a9z3 - 2a9z5 + 2a10z2 - a10z4

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 2         2
j = 0        1 
j = -2       52 
j = -4      43  
j = -6     43   
j = -8    44    
j = -10   34     
j = -12  14      
j = -14 13       
j = -16 1        
j = -181         


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, 223]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 223]]
Out[4]=   
PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 8, 19, 7], 
 
>   X[16, 9, 17, 10], X[22, 15, 9, 16], X[21, 5, 22, 4], X[5, 14, 6, 15], 
 
>   X[13, 20, 14, 21], X[8, 18, 1, 17], X[6, 20, 7, 19]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, 7, -8, -11, 4, -10}, 
 
>   {5, -1, 2, -3, -9, 8, 6, -5, 10, -4, 11, 9, -7, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(17/2)     2       4       7      8      8      7      6        3
q        - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 
            15/2    13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q       q      q      q      q
 
>   2 Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -26    -20    2     -16    -10   3    3    2       2
1 - q    - q    + --- - q    - q    + -- + -- + -- + 2 q
                   18                  8    4    2
                  q                   q    q    q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 223]][a, z]
Out[8]=   
          3    5
-2 a   3 a    a               3        7          3      3  3      5  3
---- + ---- - -- - 5 a z + 7 a  z - 2 a  z - 2 a z  + 7 a  z  + 2 a  z  - 
 z      z     z
 
     7  3      3  5    5  5
>   a  z  + 2 a  z  + a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 223]][a, z]
Out[9]=   
                             3    5
    2      4    6   2 a   3 a    a                3        7        2  2
-3 a  - 3 a  - a  + --- + ---- + -- - 9 a z - 13 a  z + 4 a  z + 6 a  z  + 
                     z     z     z
 
        4  2      6  2      8  2      10  2         3       3  3      5  3
>   16 a  z  + 5 a  z  - 3 a  z  + 2 a   z  + 10 a z  + 21 a  z  - 7 a  z  - 
 
        7  3      9  3      2  4       4  4       6  4      8  4    10  4
>   15 a  z  + 3 a  z  - 2 a  z  - 20 a  z  - 12 a  z  + 5 a  z  - a   z  - 
 
         5       3  5      5  5       7  5      9  5      2  6       4  6
>   3 a z  - 10 a  z  + 6 a  z  + 11 a  z  - 2 a  z  + 2 a  z  + 13 a  z  + 
 
       6  6      8  6      3  7    5  7      7  7    2  8      4  8      6  8
>   8 a  z  - 3 a  z  + 3 a  z  - a  z  - 4 a  z  - a  z  - 4 a  z  - 3 a  z  - 
 
     3  9    5  9
>   a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
3    5      1        1        1        3        1        4        3
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    18  7    16  6    14  6    14  5    12  5    12  4    10  4
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      4        4       4       4      3      4         2 t      2  2
>   ------ + ----- + ----- + ----- + ---- + ---- + t + --- + 2 q  t
     10  3    8  3    8  2    6  2    6      4          2
    q   t    q  t    q  t    q  t    q  t   q  t       q


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n223
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