© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11n199
L11n199
L11n201
L11n201
L11n200
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11n200

Visit L11n200's page at Knotilus!

Acknowledgement

L11n200 as Morse Link
DrawMorseLink

PD Presentation: X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X20,15,21,16 X13,18,14,19 X6,22,7,21 X22,18,9,17 X19,5,20,4 X2,9,3,10 X8,11,1,12

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

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

A2 (sl(3)) Invariant: q-26 - 2q-18 + q-16 + q-14 + 2q-12 + 3q-10 + q-8 + 2q-6 + 1 - q2

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

Kauffman Polynomial: - z2 + az - 3az3 - a2z2 - a2z6 - a3z-1 + 2a3z - 3a3z3 + 4a3z5 - 2a3z7 + a4 + a4z2 - 3a4z4 + 5a4z6 - 2a4z8 - a5z-1 - a5z + a5z3 + 3a5z5 + a5z7 - a5z9 + 8a6z2 - 18a6z4 + 16a6z6 - 4a6z8 - 6a7z3 + 4a7z5 + 2a7z7 - a7z9 + 7a8z2 - 15a8z4 + 10a8z6 - 2a8z8 + 2a9z - 7a9z3 + 5a9z5 - a9z7

Khovanov Homology:
trqj r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 2         1
j = 0        2 
j = -2       32 
j = -4      31  
j = -6     23   
j = -8    43    
j = -10   23     
j = -12  13      
j = -14 12       
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, 200]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 200]]
Out[4]=   
PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[5, 14, 6, 15], X[16, 7, 17, 8], 
 
>   X[20, 15, 21, 16], X[13, 18, 14, 19], X[6, 22, 7, 21], X[22, 18, 9, 17], 
 
>   X[19, 5, 20, 4], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, 9, -3, -7, 4, -11}, 
 
>   {10, -1, 11, -2, -6, 3, 5, -4, 8, 6, -9, -5, 7, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(17/2)     2       3       5      6      5      6      4        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
 
>   Sqrt[q]
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -26    2     -16    -14    2     3     -8   2     2
1 + q    - --- + q    + q    + --- + --- + q   + -- - q
            18                  12    10          6
           q                   q     q           q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 200]][a, z]
Out[8]=   
   3     5
  a     a             3        5        7        3      3  3      5  3
-(--) + -- + a z - 3 a  z - 2 a  z + 2 a  z + a z  - 3 a  z  - 3 a  z  + 
  z     z
 
     7  3    3  5    5  5
>   a  z  - a  z  - a  z
In[9]:=
Kauffman[Link[11, NonAlternating, 200]][a, z]
Out[9]=   
      3    5
 4   a    a             3      5        9      2    2  2    4  2      6  2
a  - -- - -- + a z + 2 a  z - a  z + 2 a  z - z  - a  z  + a  z  + 8 a  z  + 
     z    z
 
       8  2        3      3  3    5  3      7  3      9  3      4  4
>   7 a  z  - 3 a z  - 3 a  z  + a  z  - 6 a  z  - 7 a  z  - 3 a  z  - 
 
        6  4       8  4      3  5      5  5      7  5      9  5    2  6
>   18 a  z  - 15 a  z  + 4 a  z  + 3 a  z  + 4 a  z  + 5 a  z  - a  z  + 
 
       4  6       6  6       8  6      3  7    5  7      7  7    9  7
>   5 a  z  + 16 a  z  + 10 a  z  - 2 a  z  + a  z  + 2 a  z  - a  z  - 
 
       4  8      6  8      8  8    5  9    7  9
>   2 a  z  - 4 a  z  - 2 a  z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    2      1        1        1        2        1        3        2
2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     2    18  8    16  7    14  7    14  6    12  6    12  5    10  5
    q    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        4       3       2       3       3      1      3      2
>   ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + q  t
     10  4    8  4    8  3    6  3    6  2    4  2    4      2
    q   t    q  t    q  t    q  t    q  t    q  t    q  t   q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n200
L11n199
L11n199
L11n201
L11n201