© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a35
L11a35
L11a37
L11a37
L11a36
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a36

Visit L11a36's page at Knotilus!

Acknowledgement

L11a36 as Morse Link
DrawMorseLink

PD Presentation: X6172 X10,3,11,4 X18,15,19,16 X16,7,17,8 X8,17,9,18 X20,11,21,12 X22,13,5,14 X12,21,13,22 X14,19,15,20 X2536 X4,9,1,10

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

Jones Polynomial: q-27/2 - 3q-25/2 + 7q-23/2 - 10q-21/2 + 13q-19/2 - 15q-17/2 + 14q-15/2 - 13q-13/2 + 8q-11/2 - 5q-9/2 + 2q-7/2 - q-5/2

A2 (sl(3)) Invariant: - q-42 - q-40 + q-38 - 3q-36 - q-34 - 3q-30 + 3q-28 + q-26 + 4q-24 + 4q-22 + 4q-18 - q-16 + 2q-12 - q-10 + q-8

HOMFLY-PT Polynomial: - 2a5z - 3a5z3 - a5z5 - 2a7z-1 - 4a7z - 5a7z3 - 2a7z5 + 2a9z-1 - 4a9z3 - 2a9z5 + a11z-1 + 5a11z + 3a11z3 - a13z-1 - a13z

Kauffman Polynomial: - 2a5z + 3a5z3 - a5z5 - a6z2 + 4a6z4 - 2a6z6 - 2a7z-1 + 6a7z - 6a7z3 + 6a7z5 - 3a7z7 + 3a8 - 6a8z2 + 2a8z4 + 3a8z6 - 3a8z8 - 2a9z-1 + 8a9z - 7a9z3 - 4a9z5 + 5a9z7 - 3a9z9 + 5a10z2 - 21a10z4 + 17a10z6 - 5a10z8 - a10z10 + a11z-1 - a11z + a11z3 - 10a11z5 + 13a11z7 - 6a11z9 - 3a12 + 12a12z2 - 22a12z4 + 21a12z6 - 6a12z8 - a12z10 + a13z-1 - 6a13z3 + 9a13z5 + 2a13z7 - 3a13z9 - a14z2 + 8a14z6 - 4a14z8 + a15z - 5a15z3 + 8a15z5 - 3a15z7 + a16 - 3a16z2 + 3a16z4 - a16z6

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4           1
j = -6          21
j = -8         3  
j = -10        52  
j = -12       83   
j = -14      76    
j = -16     87     
j = -18    57      
j = -20   58       
j = -22  25        
j = -24 15         
j = -26 2          
j = -281           


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, 36]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 36]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 15, 19, 16], X[16, 7, 17, 8], 
 
>   X[8, 17, 9, 18], X[20, 11, 21, 12], X[22, 13, 5, 14], X[12, 21, 13, 22], 
 
>   X[14, 19, 15, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -8, 7, -9, 3, -4, 5, -3, 
 
>    9, -6, 8, -7}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(27/2)     3       7      10      13      15      14      13       8
q        - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 
            25/2    23/2    21/2    19/2    17/2    15/2    13/2    11/2
           q       q       q       q       q       q       q       q
 
     5      2      -(5/2)
>   ---- + ---- - q
     9/2    7/2
    q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    -40    -38    3     -34    3     3     -26    4     4     4     -16
-q    - q    + q    - --- - q    - --- + --- + q    + --- + --- + --- - q    + 
                       36           30    28           24    22    18
                      q            q     q            q     q     q
 
     2     -10    -8
>   --- - q    + q
     12
    q
In[8]:=
HOMFLYPT[Link[11, Alternating, 36]][a, z]
Out[8]=   
    7      9    11    13
-2 a    2 a    a     a        5        7        11      13        5  3
----- + ---- + --- - --- - 2 a  z - 4 a  z + 5 a   z - a   z - 3 a  z  - 
  z      z      z     z
 
       7  3      9  3      11  3    5  5      7  5      9  5
>   5 a  z  - 4 a  z  + 3 a   z  - a  z  - 2 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 36]][a, z]
Out[9]=   
                        7      9    11    13
   8      12    16   2 a    2 a    a     a        5        7        9
3 a  - 3 a   + a   - ---- - ---- + --- + --- - 2 a  z + 6 a  z + 8 a  z - 
                      z      z      z     z
 
     11      15      6  2      8  2      10  2       12  2    14  2
>   a   z + a   z - a  z  - 6 a  z  + 5 a   z  + 12 a   z  - a   z  - 
 
       16  2      5  3      7  3      9  3    11  3      13  3      15  3
>   3 a   z  + 3 a  z  - 6 a  z  - 7 a  z  + a   z  - 6 a   z  - 5 a   z  + 
 
       6  4      8  4       10  4       12  4      16  4    5  5      7  5
>   4 a  z  + 2 a  z  - 21 a   z  - 22 a   z  + 3 a   z  - a  z  + 6 a  z  - 
 
       9  5       11  5      13  5      15  5      6  6      8  6       10  6
>   4 a  z  - 10 a   z  + 9 a   z  + 8 a   z  - 2 a  z  + 3 a  z  + 17 a   z  + 
 
        12  6      14  6    16  6      7  7      9  7       11  7      13  7
>   21 a   z  + 8 a   z  - a   z  - 3 a  z  + 5 a  z  + 13 a   z  + 2 a   z  - 
 
       15  7      8  8      10  8      12  8      14  8      9  9      11  9
>   3 a   z  - 3 a  z  - 5 a   z  - 6 a   z  - 4 a   z  - 3 a  z  - 6 a   z  - 
 
       13  9    10  10    12  10
>   3 a   z  - a   z   - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -6    -4      1         2         1        5        2        5        5
q   + q   + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 
             28  11    26  10    24  10    24  9    22  9    22  8    20  8
            q   t     q   t     q   t     q   t    q   t    q   t    q   t
 
      8        5        7        8        7        7        6        8
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     20  7    18  7    18  6    16  6    16  5    14  5    14  4    12  4
    q   t    q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        5        2        3      2
>   ------ + ------ + ------ + ----- + ----
     12  3    10  3    10  2    8  2    6
    q   t    q   t    q   t    q  t    q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a36
L11a35
L11a35
L11a37
L11a37