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L11a230

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Acknowledgement

L11a230 as Morse Link
DrawMorseLink

PD Presentation: X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,13,19,14 X14,17,15,18 X6718 X4,15,5,16 X22,20,7,19 X20,10,21,9 X10,22,11,21

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

Jones Polynomial: - q-19/2 + 3q-17/2 - 7q-15/2 + 12q-13/2 - 16q-11/2 + 18q-9/2 - 19q-7/2 + 15q-5/2 - 12q-3/2 + 7q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant: q-28 - q-26 + 2q-24 - 3q-20 + 3q-18 - 2q-16 + 4q-14 + 3q-12 + q-10 + 5q-8 - 3q-6 + 2q-4 - q-2 - 2 + q2 - q4

HOMFLY-PT Polynomial: az-1 + 3az + 3az3 + az5 - 3a3z-1 - 7a3z - 7a3z3 - 4a3z5 - a3z7 + 2a5z-1 - 2a5z - 6a5z3 - 4a5z5 - a5z7 + 3a7z + 3a7z3 + a7z5

Kauffman Polynomial: 1 - 3z2 + 3z4 - z6 - az-1 + 3az - 7az3 + 8az5 - 3az7 + 3a2 - 8a2z2 + 4a2z4 + 6a2z6 - 4a2z8 - 3a3z-1 + 10a3z - 18a3z3 + 18a3z5 - 2a3z7 - 3a3z9 + 3a4 - 2a4z2 - 10a4z4 + 21a4z6 - 9a4z8 - a4z10 - 2a5z-1 + 4a5z - 10a5z3 + 8a5z5 + 6a5z7 - 7a5z9 + 9a6z2 - 24a6z4 + 26a6z6 - 11a6z8 - a6z10 - a7z - 4a7z3 + 6a7z5 - 4a7z9 + 4a8z2 - 8a8z4 + 9a8z6 - 6a8z8 + a9z - 3a9z3 + 7a9z5 - 5a9z7 - 2a10z2 + 5a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3
j = 4           1
j = 2          2 
j = 0         51 
j = -2        72  
j = -4       96   
j = -6      106    
j = -8     89     
j = -10    810      
j = -12   48       
j = -14  38        
j = -16 15         
j = -18 2          
j = -201           


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, Alternating, 230]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 230]]
Out[4]=   
PD[X[8, 1, 9, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[16, 5, 17, 6], 
 
>   X[18, 13, 19, 14], X[14, 17, 15, 18], X[6, 7, 1, 8], X[4, 15, 5, 16], 
 
>   X[22, 20, 7, 19], X[20, 10, 21, 9], X[10, 22, 11, 21]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -2, 3, -8, 4, -7}, 
 
>   {7, -1, 10, -11, 2, -3, 5, -6, 8, -4, 6, -5, 9, -10, 11, -9}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(19/2)     3       7      12      16      18     19     15     12
-q        + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 
             17/2    15/2    13/2    11/2    9/2    7/2    5/2    3/2
            q       q       q       q       q      q      q      q
 
       7                   3/2
>   ------- - 3 Sqrt[q] + q
    Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -28    -26    2     3     3     2     4     3     -10   5    3    2
-2 + q    - q    + --- - --- + --- - --- + --- + --- + q    + -- - -- + -- - 
                    24    20    18    16    14    12           8    6    4
                   q     q     q     q     q     q            q    q    q
 
     -2    2    4
>   q   + q  - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 230]][a, z]
Out[8]=   
       3      5
a   3 a    2 a               3        5        7          3      3  3
- - ---- + ---- + 3 a z - 7 a  z - 2 a  z + 3 a  z + 3 a z  - 7 a  z  - 
z    z      z
 
       5  3      7  3      5      3  5      5  5    7  5    3  7    5  7
>   6 a  z  + 3 a  z  + a z  - 4 a  z  - 4 a  z  + a  z  - a  z  - a  z
In[9]:=
Kauffman[Link[11, Alternating, 230]][a, z]
Out[9]=   
                         3      5
       2      4   a   3 a    2 a                3        5      7      9
1 + 3 a  + 3 a  - - - ---- - ---- + 3 a z + 10 a  z + 4 a  z - a  z + a  z - 
                  z    z      z
 
     11        2      2  2      4  2      6  2      8  2      10  2        3
>   a   z - 3 z  - 8 a  z  - 2 a  z  + 9 a  z  + 4 a  z  - 2 a   z  - 7 a z  - 
 
        3  3       5  3      7  3      9  3      11  3      4      2  4
>   18 a  z  - 10 a  z  - 4 a  z  - 3 a  z  + 2 a   z  + 3 z  + 4 a  z  - 
 
        4  4       6  4      8  4      10  4        5       3  5      5  5
>   10 a  z  - 24 a  z  - 8 a  z  + 5 a   z  + 8 a z  + 18 a  z  + 8 a  z  + 
 
       7  5      9  5    11  5    6      2  6       4  6       6  6      8  6
>   6 a  z  + 7 a  z  - a   z  - z  + 6 a  z  + 21 a  z  + 26 a  z  + 9 a  z  - 
 
       10  6        7      3  7      5  7      9  7      2  8      4  8
>   3 a   z  - 3 a z  - 2 a  z  + 6 a  z  - 5 a  z  - 4 a  z  - 9 a  z  - 
 
        6  8      8  8      3  9      5  9      7  9    4  10    6  10
>   11 a  z  - 6 a  z  - 3 a  z  - 7 a  z  - 4 a  z  - a  z   - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
6    7      1        2        1        5        3        8        4
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
 4    2    20  8    18  7    16  7    16  6    14  6    14  5    12  5
q    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      8        8        10       8       9      10      6      9
>   ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 t + 
     12  4    10  4    10  3    8  3    8  2    6  2    6      4
    q   t    q   t    q   t    q  t    q  t    q  t    q  t   q  t
 
    2 t    2      2  2    4  3
>   --- + t  + 2 q  t  + q  t
     2
    q


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a230
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