© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a299
L11a299
L11a301
L11a301
L11a300
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a300

Visit L11a300's page at Knotilus!

Acknowledgement

L11a300 as Morse Link
DrawMorseLink

PD Presentation: X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,15,21,16 X8,14,1,13 X4,20,5,19 X18,8,19,7 X16,6,17,5 X6,18,7,17 X14,21,15,22

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

Jones Polynomial: - q-5/2 + 3q-3/2 - 6q-1/2 + 10q1/2 - 14q3/2 + 16q5/2 - 18q7/2 + 15q9/2 - 12q11/2 + 8q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant: q-8 - q-4 + q-2 - 3 + q2 + 2q4 - q6 + 5q8 - q10 + 3q12 + q14 - q16 + 3q18 - 2q20 + q22 + q24 - q26

HOMFLY-PT Polynomial: a-7z + a-7z3 - a-5z-1 - 3a-5z - 5a-5z3 - 2a-5z5 + a-3z-1 + 6a-3z + 7a-3z3 + 4a-3z5 + a-3z7 - 4a-1z - 6a-1z3 - 2a-1z5 + 2az + az3

Kauffman Polynomial: - a-10z4 + 2a-9z3 - 4a-9z5 - 2a-8z2 + 8a-8z4 - 8a-8z6 + 2a-7z - 7a-7z3 + 13a-7z5 - 10a-7z7 - 2a-6z2 + 5a-6z4 + 6a-6z6 - 8a-6z8 - a-5z-1 + 7a-5z - 23a-5z3 + 30a-5z5 - 7a-5z7 - 4a-5z9 + a-4 + 3a-4z2 - 21a-4z4 + 33a-4z6 - 12a-4z8 - a-4z10 - a-3z-1 + 6a-3z - 19a-3z3 + 11a-3z5 + 11a-3z7 - 7a-3z9 + 9a-2z2 - 32a-2z4 + 31a-2z6 - 7a-2z8 - a-2z10 + 3a-1z - 10a-1z3 + 2a-1z5 + 7a-1z7 - 3a-1z9 + 6z2 - 15z4 + 12z6 - 3z8 + 2az - 5az3 + 4az5 - az7

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 18           1
j = 16          3 
j = 14         51 
j = 12        73  
j = 10       96   
j = 8      96    
j = 6     79     
j = 4    79      
j = 2   48       
j = 0  26        
j = -2 14         
j = -4 2          
j = -61           


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, 300]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 300]]
Out[4]=   
PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], 
 
>   X[20, 15, 21, 16], X[8, 14, 1, 13], X[4, 20, 5, 19], X[18, 8, 19, 7], 
 
>   X[16, 6, 17, 5], X[6, 18, 7, 17], X[14, 21, 15, 22]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, 2, -7, 9, -10, 8, -6}, 
 
>   {4, -1, 3, -2, 6, -11, 5, -9, 10, -8, 7, -5, 11, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(5/2)    3        6                       3/2       5/2       7/2
-q       + ---- - ------- + 10 Sqrt[q] - 14 q    + 16 q    - 18 q    + 
            3/2   Sqrt[q]
           q
 
        9/2       11/2      13/2      15/2    17/2
>   15 q    - 12 q     + 8 q     - 4 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -8    -4    -2    2      4    6      8    10      12    14    16
-3 + q   - q   + q   + q  + 2 q  - q  + 5 q  - q   + 3 q   + q   - q   + 
 
       18      20    22    24    26
>   3 q   - 2 q   + q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 300]][a, z]
Out[8]=   
                                                 3      3      3      3
   1       1     z    3 z   6 z   4 z           z    5 z    7 z    6 z
-(----) + ---- + -- - --- + --- - --- + 2 a z + -- - ---- + ---- - ---- + 
   5       3      7    5     3     a             7     5      3     a
  a  z    a  z   a    a     a                   a     a      a
 
              5      5      5    7
       3   2 z    4 z    2 z    z
>   a z  - ---- + ---- - ---- + --
             5      3     a      3
            a      a            a
In[9]:=
Kauffman[Link[11, Alternating, 300]][a, z]
Out[9]=   
                                                              2      2      2
 -4    1      1     2 z   7 z   6 z   3 z              2   2 z    2 z    3 z
a   - ---- - ---- + --- + --- + --- + --- + 2 a z + 6 z  - ---- - ---- + ---- + 
       5      3      7     5     3     a                     8      6      4
      a  z   a  z   a     a     a                           a      a      a
 
       2      3      3       3       3       3                     4       4
    9 z    2 z    7 z    23 z    19 z    10 z         3       4   z     8 z
>   ---- + ---- - ---- - ----- - ----- - ----- - 5 a z  - 15 z  - --- + ---- + 
      2      9      7      5       3       a                       10     8
     a      a      a      a       a                               a      a
 
       4       4       4      5       5       5       5      5
    5 z    21 z    32 z    4 z    13 z    30 z    11 z    2 z         5
>   ---- - ----- - ----- - ---- + ----- + ----- + ----- + ---- + 4 a z  + 
      6      4       2       9      7       5       3      a
     a      a       a       a      a       a       a
 
               6      6       6       6       7      7       7      7
        6   8 z    6 z    33 z    31 z    10 z    7 z    11 z    7 z       7
>   12 z  - ---- + ---- + ----- + ----- - ----- - ---- + ----- + ---- - a z  - 
              8      6      4       2       7       5      3      a
             a      a      a       a       a       a      a
 
              8       8      8      9      9      9    10    10
       8   8 z    12 z    7 z    4 z    7 z    3 z    z     z
>   3 z  - ---- - ----- - ---- - ---- - ---- - ---- - --- - ---
             6      4       2      5      3     a      4     2
            a      a       a      a      a            a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
                                                          2
   2      4     1       2       1     2      4     6   4 q       4        6
8 q  + 7 q  + ----- + ----- + ----- + -- + ----- + - + ---- + 9 q  t + 7 q  t + 
               6  4    4  3    2  3    2    2  2   t    t
              q  t    q  t    q  t    t    q  t
 
       6  2      8  2      8  3      10  3      10  4      12  4      12  5
>   9 q  t  + 9 q  t  + 6 q  t  + 9 q   t  + 6 q   t  + 7 q   t  + 3 q   t  + 
 
       14  5    14  6      16  6    18  7
>   5 q   t  + q   t  + 3 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a300
L11a299
L11a299
L11a301
L11a301