© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a72
L11a72
L11a74
L11a74
L11a73
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a73

Visit L11a73's page at Knotilus!

Acknowledgement

L11a73 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-27/2 - 2q-25/2 + 6q-23/2 - 10q-21/2 + 13q-19/2 - 16q-17/2 + 16q-15/2 - 15q-13/2 + 10q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant: - q-42 - 2q-40 - q-38 - 4q-36 - q-34 + 3q-32 + 5q-28 + q-26 + q-24 + 3q-22 - q-20 + 5q-18 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial: - 2a5z3 - a5z5 - a7z-1 - 7a7z - 9a7z3 - 3a7z5 - a9z-1 - a9z - 4a9z3 - 2a9z5 + 4a11z-1 + 7a11z + 3a11z3 - 2a13z-1 - a13z

Kauffman Polynomial: 2a5z3 - a5z5 + 5a6z4 - 3a6z6 - a7z-1 + 7a7z - 15a7z3 + 15a7z5 - 6a7z7 + a8 - 9a8z4 + 12a8z6 - 6a8z8 + a9z-1 - 2a9z - 4a9z3 + a9z5 + 4a9z7 - 4a9z9 - 5a10 + 22a10z2 - 41a10z4 + 30a10z6 - 9a10z8 - a10z10 + 4a11z-1 - 17a11z + 31a11z3 - 29a11z5 + 18a11z7 - 7a11z9 - 6a12 + 25a12z2 - 32a12z4 + 22a12z6 - 6a12z8 - a12z10 + 2a13z-1 - 9a13z + 16a13z3 - 9a13z5 + 6a13z7 - 3a13z9 + a14 - 2a14z2 - a14z4 + 6a14z6 - 3a14z8 - a15z - 2a15z3 + 5a15z5 - 2a15z7 + 2a16 - 5a16z2 + 4a16z4 - a16z6

Khovanov Homology:
trqj r = -11r = -10r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -4           1
j = -6          31
j = -8         4  
j = -10        63  
j = -12       94   
j = -14      87    
j = -16     88     
j = -18    58      
j = -20   58       
j = -22  15        
j = -24 15         
j = -26 1          
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, 73]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 73]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[20, 13, 21, 14], X[16, 7, 17, 8], 
 
>   X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[22, 15, 5, 16], 
 
>   X[14, 21, 15, 22], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, 3, -9, 8, -4, 7, -6, 
 
>    5, -3, 9, -8}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(27/2)     2       6      10      13      16      16      15      10
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
 
     7      3      -(5/2)
>   ---- + ---- - q
     9/2    7/2
    q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -42    2     -38    4     -34    3     5     -26    -24    3     -20    5
-q    - --- - q    - --- - q    + --- + --- + q    + q    + --- - q    + --- + 
         40           36           32    28                  22           18
        q            q            q     q                   q            q
 
     2     2     -8
>   --- - --- + q
     12    10
    q     q
In[8]:=
HOMFLYPT[Link[11, Alternating, 73]][a, z]
Out[8]=   
   7     9      11      13
  a     a    4 a     2 a        7      9        11      13        5  3
-(--) - -- + ----- - ----- - 7 a  z - a  z + 7 a   z - a   z - 2 a  z  - 
  z     z      z       z
 
       7  3      9  3      11  3    5  5      7  5      9  5
>   9 a  z  - 4 a  z  + 3 a   z  - a  z  - 3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[11, Alternating, 73]][a, z]
Out[9]=   
                                    7    9      11      13
 8      10      12    14      16   a    a    4 a     2 a        7        9
a  - 5 a   - 6 a   + a   + 2 a   - -- + -- + ----- + ----- + 7 a  z - 2 a  z - 
                                   z    z      z       z
 
        11        13      15         10  2       12  2      14  2      16  2
>   17 a   z - 9 a   z - a   z + 22 a   z  + 25 a   z  - 2 a   z  - 5 a   z  + 
 
       5  3       7  3      9  3       11  3       13  3      15  3      6  4
>   2 a  z  - 15 a  z  - 4 a  z  + 31 a   z  + 16 a   z  - 2 a   z  + 5 a  z  - 
 
       8  4       10  4       12  4    14  4      16  4    5  5       7  5
>   9 a  z  - 41 a   z  - 32 a   z  - a   z  + 4 a   z  - a  z  + 15 a  z  + 
 
     9  5       11  5      13  5      15  5      6  6       8  6       10  6
>   a  z  - 29 a   z  - 9 a   z  + 5 a   z  - 3 a  z  + 12 a  z  + 30 a   z  + 
 
        12  6      14  6    16  6      7  7      9  7       11  7      13  7
>   22 a   z  + 6 a   z  - a   z  - 6 a  z  + 4 a  z  + 18 a   z  + 6 a   z  - 
 
       15  7      8  8      10  8      12  8      14  8      9  9      11  9
>   2 a   z  - 6 a  z  - 9 a   z  - 6 a   z  - 3 a   z  - 4 a  z  - 7 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         1         1        5        1        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        8        8        8        8        7        9
>   ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
     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
 
      4        6        3        4      3
>   ------ + ------ + ------ + ----- + ----
     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 L11a73
L11a72
L11a72
L11a74
L11a74