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The 2-Component Link

L11n2

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Acknowledgement

L11n2 as Morse Link
DrawMorseLink

PD Presentation: X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X2,14,3,13

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

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

A2 (sl(3)) Invariant: - q-24 - q-22 - 2q-18 + q-16 + 2q-14 + 3q-10 + q-8 + 2q-6 + q-4 - q-2 + 2 - q2 + q4 + 2q6

HOMFLY-PT Polynomial: - a-1z-1 - 2a-1z + 2az-1 + 5az + 3az3 - 3a3z-1 - 6a3z - 3a3z3 - a3z5 + 3a5z-1 + 4a5z + 2a5z3 - a7z-1 - a7z

Kauffman Polynomial: - a-1z-1 + 4a-1z - 3a-1z3 + 4z2 - 2z4 - z6 - 2az-1 + 11az - 17az3 + 9az5 - 3az7 - 2a2 + 11a2z2 - 18a2z4 + 10a2z6 - 3a2z8 - 3a3z-1 + 14a3z - 24a3z3 + 14a3z5 - 2a3z7 - a3z9 + 8a4z2 - 20a4z4 + 17a4z6 - 5a4z8 - 3a5z-1 + 10a5z - 16a5z3 + 12a5z5 - a5z7 - a5z9 + 2a6 - 3a6z2 + 5a6z6 - 2a6z8 - a7z-1 + 3a7z - 6a7z3 + 7a7z5 - 2a7z7 + a8 - 4a8z2 + 4a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2
j = 4         2
j = 2        2 
j = 0       42 
j = -2      54  
j = -4     42   
j = -6    35    
j = -8   34     
j = -10  13      
j = -12 13       
j = -14 1        
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, NonAlternating, 2]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 2]]
Out[4]=   
PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[15, 1, 16, 4], X[5, 10, 6, 11], 
 
>   X[3, 8, 4, 9], X[11, 20, 12, 21], X[17, 5, 18, 22], X[21, 19, 22, 18], 
 
>   X[19, 12, 20, 13], X[9, 16, 10, 17], X[2, 14, 3, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -6, 9, 11, -2, -3, 10, -7, 8, 
 
>    -9, 6, -8, 7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     2       4      6      7      9      7        6
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 4 Sqrt[q] - 
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
 
       3/2
>   2 q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -24    -22    2     -16    2     3     -8   2     -4    -2    2    4      6
2 - q    - q    - --- + q    + --- + --- + q   + -- + q   - q   - q  + q  + 2 q
                   18           14    10          6
                  q            q     q           q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 2]][a, z]
Out[8]=   
                  3      5    7
   1     2 a   3 a    3 a    a    2 z              3        5      7
-(---) + --- - ---- + ---- - -- - --- + 5 a z - 6 a  z + 4 a  z - a  z + 
  a z     z     z      z     z     a
 
         3      3  3      5  3    3  5
>   3 a z  - 3 a  z  + 2 a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 2]][a, z]
Out[9]=   
                                   3      5    7
    2      6    8    1    2 a   3 a    3 a    a    4 z                3
-2 a  + 2 a  + a  - --- - --- - ---- - ---- - -- + --- + 11 a z + 14 a  z + 
                    a z    z     z      z     z     a
 
                                                                          3
        5        7        2       2  2      4  2      6  2      8  2   3 z
>   10 a  z + 3 a  z + 4 z  + 11 a  z  + 8 a  z  - 3 a  z  - 4 a  z  - ---- - 
                                                                        a
 
          3       3  3       5  3      7  3      4       2  4       4  4
>   17 a z  - 24 a  z  - 16 a  z  - 6 a  z  - 2 z  - 18 a  z  - 20 a  z  + 
 
       8  4        5       3  5       5  5      7  5    6       2  6
>   4 a  z  + 9 a z  + 14 a  z  + 12 a  z  + 7 a  z  - z  + 10 a  z  + 
 
        4  6      6  6    8  6        7      3  7    5  7      7  7      2  8
>   17 a  z  + 5 a  z  - a  z  - 3 a z  - 2 a  z  - a  z  - 2 a  z  - 3 a  z  - 
 
       4  8      6  8    3  9    5  9
>   5 a  z  - 2 a  z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    4      1        1        1        3        1        3        3       4
4 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
     2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
    q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      3       5       4      2      5              2        4  2
>   ----- + ----- + ----- + ---- + ---- + 2 t + 2 q  t + 2 q  t
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n2
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