© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a360
L11a360
L11a362
L11a362
L11a361
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a361

Visit L11a361's page at Knotilus!

Acknowledgement

L11a361 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q1/2 + 2q3/2 - 5q5/2 + 8q7/2 - 11q9/2 + 13q11/2 - 14q13/2 + 12q15/2 - 10q17/2 + 6q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant: q2 + q6 + q8 - 2q10 + 2q12 - 2q14 + 2q16 + 2q18 + 4q22 - q24 + 2q26 - q30 + q32 - q34

HOMFLY-PT Polynomial: 2a-9z + 3a-9z3 + a-9z5 - a-7z-1 - 3a-7z - 5a-7z3 - 4a-7z5 - a-7z7 + a-5z-1 - 4a-5z3 - 4a-5z5 - a-5z7 + 4a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial: a-14z2 - a-14z4 - a-13z + 3a-13z3 - 3a-13z5 - 2a-12z2 + 5a-12z4 - 5a-12z6 - 4a-11z3 + 7a-11z5 - 6a-11z7 - 3a-10z2 + 2a-10z4 + 4a-10z6 - 5a-10z8 + 2a-9z - 3a-9z3 + 6a-9z5 - 3a-9z9 - 2a-8z2 + a-8z4 + 9a-8z6 - 4a-8z8 - a-8z10 + a-7z-1 - 5a-7z + 9a-7z3 - 9a-7z5 + 12a-7z7 - 5a-7z9 - a-6 - 4a-6z4 + 8a-6z6 - a-6z8 - a-6z10 + a-5z-1 - 2a-5z - 3a-5z3 + 5a-5z7 - 2a-5z9 + 2a-4z2 - 9a-4z4 + 8a-4z6 - 2a-4z8 + 4a-3z - 8a-3z3 + 5a-3z5 - a-3z7

Khovanov Homology:
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 24           1
j = 22          2 
j = 20         41 
j = 18        62  
j = 16       75   
j = 14      75    
j = 12     67     
j = 10    57      
j = 8   36       
j = 6  25        
j = 4 14         
j = 2 1          
j = 01           


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, 361]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 361]]
Out[4]=   
PD[X[12, 1, 13, 2], X[14, 4, 15, 3], X[22, 14, 11, 13], X[16, 6, 17, 5], 
 
>   X[2, 11, 3, 12], X[4, 16, 5, 15], X[6, 22, 7, 21], X[20, 10, 21, 9], 
 
>   X[18, 8, 19, 7], X[8, 20, 9, 19], X[10, 18, 1, 17]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -5, 2, -6, 4, -7, 9, -10, 8, -11}, 
 
>   {5, -1, 3, -2, 6, -4, 11, -9, 10, -8, 7, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
              3/2      5/2      7/2       9/2       11/2       13/2
-Sqrt[q] + 2 q    - 5 q    + 8 q    - 11 q    + 13 q     - 14 q     + 
 
        15/2       17/2      19/2      21/2    23/2
>   12 q     - 10 q     + 6 q     - 3 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
 2    6    8      10      12      14      16      18      22    24      26
q  + q  + q  - 2 q   + 2 q   - 2 q   + 2 q   + 2 q   + 4 q   - q   + 2 q   - 
 
     30    32    34
>   q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 361]][a, z]
Out[8]=   
                                      3      3      3      3    5      5
   1       1     2 z   3 z   4 z   3 z    5 z    4 z    4 z    z    4 z
-(----) + ---- + --- - --- + --- + ---- - ---- - ---- + ---- + -- - ---- - 
   7       5      9     7     3      9      7      5      3     9     7
  a  z    a  z   a     a     a      a      a      a      a     a     a
 
       5    5    7    7
    4 z    z    z    z
>   ---- + -- - -- - --
      5     3    7    5
     a     a    a    a
In[9]:=
Kauffman[Link[11, Alternating, 361]][a, z]
Out[9]=   
                                                    2       2      2      2
  -6    1      1      z    2 z   5 z   2 z   4 z   z     2 z    3 z    2 z
-a   + ---- + ---- - --- + --- - --- - --- + --- + --- - ---- - ---- - ---- + 
        7      5      13    9     7     5     3     14    12     10      8
       a  z   a  z   a     a     a     a     a     a     a      a       a
 
       2      3      3      3      3      3      3    4       4      4    4
    2 z    3 z    4 z    3 z    9 z    3 z    8 z    z     5 z    2 z    z
>   ---- + ---- - ---- - ---- + ---- - ---- - ---- - --- + ---- + ---- + -- - 
      4     13     11      9      7      5      3     14    12     10     8
     a     a      a       a      a      a      a     a     a      a      a
 
       4      4      5      5      5      5      5      6      6      6
    4 z    9 z    3 z    7 z    6 z    9 z    5 z    5 z    4 z    9 z
>   ---- - ---- - ---- + ---- + ---- - ---- + ---- - ---- + ---- + ---- + 
      6      4     13     11      9      7      3     12     10      8
     a      a     a      a       a      a      a     a      a       a
 
       6      6      7       7      7    7      8      8    8      8      9
    8 z    8 z    6 z    12 z    5 z    z    5 z    4 z    z    2 z    3 z
>   ---- + ---- - ---- + ----- + ---- - -- - ---- - ---- - -- - ---- - ---- - 
      6      4     11      7       5     3    10      8     6     4      9
     a      a     a       a       a     a    a       a     a     a      a
 
       9      9    10    10
    5 z    2 z    z     z
>   ---- - ---- - --- - ---
      7      5     8     6
     a      a     a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
                     2    4
   4      6    -2   q    q       6        8        8  2      10  2      10  3
4 q  + 2 q  + t   + -- + -- + 5 q  t + 3 q  t + 6 q  t  + 5 q   t  + 7 q   t  + 
                    t    t
 
       12  3      12  4      14  4      14  5      16  5      16  6
>   6 q   t  + 7 q   t  + 7 q   t  + 5 q   t  + 7 q   t  + 5 q   t  + 
 
       18  6      18  7      20  7    20  8      22  8    24  9
>   6 q   t  + 2 q   t  + 4 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a361
L11a360
L11a360
L11a362
L11a362