© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L9a6
L9a6
L9a8
L9a8
L9a7
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L9a7

Visit L9a7's page at Knotilus!

Acknowledgement

L9a7 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: q-21/2 - 2q-19/2 + 4q-17/2 - 5q-15/2 + 5q-13/2 - 7q-11/2 + 5q-9/2 - 4q-7/2 + 2q-5/2 - q-3/2

A2 (sl(3)) Invariant: - q-34 - 2q-32 - q-28 - q-26 + 2q-24 + q-22 + 3q-20 + 3q-18 + 2q-16 + 2q-14 - q-12 + q-10 + q-8 - q-6 + q-4

HOMFLY-PT Polynomial: - a3z - a3z3 - a5z-1 - 2a5z - 2a5z3 - 2a7z - 2a7z3 + 2a9z-1 + 3a9z - a11z-1

Kauffman Polynomial: a3z - a3z3 + a4z2 - 2a4z4 + a5z-1 - 3a5z + 3a5z3 - 3a5z5 - a6 + a6z2 + 2a6z4 - 3a6z6 - a7z - 2a7z3 + 5a7z5 - 3a7z7 + 3a8 - 10a8z2 + 13a8z4 - 3a8z6 - a8z8 - 2a9z-1 + 5a9z - 12a9z3 + 15a9z5 - 5a9z7 + 5a10 - 15a10z2 + 13a10z4 - a10z6 - a10z8 - a11z-1 + 2a11z - 6a11z3 + 7a11z5 - 2a11z7 + 2a12 - 5a12z2 + 4a12z4 - a12z6

Khovanov Homology:
trqj r = -9r = -8r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0
j = -2         1
j = -4        21
j = -6       2  
j = -8      32  
j = -10     42   
j = -12    24    
j = -14   33     
j = -16  12      
j = -18 13       
j = -20 1        
j = -221         


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[9, Alternating, 7]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[9, Alternating, 7]]
Out[4]=   
PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 13, 17, 14], X[14, 7, 15, 8], 
 
>   X[8, 15, 9, 16], X[18, 11, 5, 12], X[12, 17, 13, 18], X[2, 5, 3, 6], 
 
>   X[4, 9, 1, 10]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -8, 2, -9}, {8, -1, 4, -5, 9, -2, 6, -7, 3, -4, 5, -3, 7, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(21/2)     2       4       5       5       7      5      4      2      -(3/2)
q        - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - q
            19/2    17/2    15/2    13/2    11/2    9/2    7/2    5/2
           q       q       q       q       q       q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -34    2     -28    -26    2     -22    3     3     2     2     -12    -10
-q    - --- - q    - q    + --- + q    + --- + --- + --- + --- - q    + q    + 
         32                  24           20    18    16    14
        q                   q            q     q     q     q
 
     -8    -6    -4
>   q   - q   + q
In[8]:=
HOMFLYPT[Link[9, Alternating, 7]][a, z]
Out[8]=   
   5       9    11
  a     2 a    a      3        5        7        9      3  3      5  3      7  3
-(--) + ---- - --- - a  z - 2 a  z - 2 a  z + 3 a  z - a  z  - 2 a  z  - 2 a  z
  z      z      z
In[9]:=
Kauffman[Link[9, Alternating, 7]][a, z]
Out[9]=   
                              5      9    11
  6      8      10      12   a    2 a    a      3        5      7        9
-a  + 3 a  + 5 a   + 2 a   + -- - ---- - --- + a  z - 3 a  z - a  z + 5 a  z + 
                             z     z      z
 
       11      4  2    6  2       8  2       10  2      12  2    3  3
>   2 a   z + a  z  + a  z  - 10 a  z  - 15 a   z  - 5 a   z  - a  z  + 
 
       5  3      7  3       9  3      11  3      4  4      6  4       8  4
>   3 a  z  - 2 a  z  - 12 a  z  - 6 a   z  - 2 a  z  + 2 a  z  + 13 a  z  + 
 
        10  4      12  4      5  5      7  5       9  5      11  5      6  6
>   13 a   z  + 4 a   z  - 3 a  z  + 5 a  z  + 15 a  z  + 7 a   z  - 3 a  z  - 
 
       8  6    10  6    12  6      7  7      9  7      11  7    8  8    10  8
>   3 a  z  - a   z  - a   z  - 3 a  z  - 5 a  z  - 2 a   z  - a  z  - a   z
In[10]:=
Kh[L][q, t]
Out[10]=   
 -4    -2     1        1        1        3        1        2        3
q   + q   + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 
             22  9    20  8    18  8    18  7    16  7    16  6    14  6
            q   t    q   t    q   t    q   t    q   t    q   t    q   t
 
      3        2        4        4        2        3       2       2      2
>   ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----
     14  5    12  5    12  4    10  4    10  3    8  3    8  2    6  2    4
    q   t    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 L9a7
L9a6
L9a6
L9a8
L9a8