© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11n97
L11n97
L11n99
L11n99
L11n98
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11n98

Visit L11n98's page at Knotilus!

Acknowledgement

L11n98 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X13,21,14,20 X7,17,8,16 X9,19,10,18 X17,9,18,8 X19,11,20,10 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12

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

Jones Polynomial: - q-3/2 - q1/2 - q3/2 + 2q5/2 - 4q7/2 + 4q9/2 - 4q11/2 + 4q13/2 - 2q15/2 + q17/2

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

HOMFLY-PT Polynomial: a-7z + a-7z3 + 3a-5z-1 + 7a-5z + 3a-5z3 - 7a-3z-1 - 20a-3z - 20a-3z3 - 8a-3z5 - a-3z7 + 4a-1z-1 + 10a-1z + 6a-1z3 + a-1z5

Kauffman Polynomial: - a-10 + 2a-10z2 - a-10z4 + 3a-9z3 - 2a-9z5 + a-8z2 + 2a-8z4 - 2a-8z6 + a-7z - a-7z7 + a-6z4 - 2a-6z6 + 3a-5z-1 - 8a-5z + 6a-5z3 - 3a-5z5 - 7a-4 + 24a-4z2 - 26a-4z4 + 9a-4z6 - a-4z8 + 7a-3z-1 - 29a-3z + 46a-3z3 - 33a-3z5 + 10a-3z7 - a-3z9 - 7a-2 + 23a-2z2 - 24a-2z4 + 9a-2z6 - a-2z8 + 4a-1z-1 - 20a-1z + 37a-1z3 - 28a-1z5 + 9a-1z7 - a-1z9

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7
j = 18           1
j = 16          1 
j = 14         31 
j = 12        11  
j = 10       33   
j = 8     121    
j = 6     13     
j = 4   122      
j = 2    2       
j = 0  1         
j = -21           
j = -41           


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, NonAlternating, 98]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, NonAlternating, 98]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 21, 14, 20], X[7, 17, 8, 16], 
 
>   X[9, 19, 10, 18], X[17, 9, 18, 8], X[19, 11, 20, 10], X[15, 5, 16, 22], 
 
>   X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, -4, 6, -5, 7, 11, -2, -3, 9, -8, 4, -6, 5, 
 
>    -7, 3, -9, 8}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(3/2)              3/2      5/2      7/2      9/2      11/2      13/2
-q       - Sqrt[q] - q    + 2 q    - 4 q    + 4 q    - 4 q     + 4 q     - 
 
       15/2    17/2
>   2 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -4   2       2      4    6      8    10    12      16    18      20
3 + q   + -- + 3 q  + 4 q  + q  + 2 q  + q   - q   - 3 q   - q   - 2 q   - 
           2
          q
 
     22    24    26
>   q   + q   - q
In[8]:=
HOMFLYPT[Link[11, NonAlternating, 98]][a, z]
Out[8]=   
                                              3      3       3      3      5
 3      7      4    z    7 z   20 z   10 z   z    3 z    20 z    6 z    8 z
---- - ---- + --- + -- + --- - ---- + ---- + -- + ---- - ----- + ---- - ---- + 
 5      3     a z    7    5      3     a      7     5      3      a       3
a  z   a  z         a    a      a            a     a      a              a
 
     5    7
    z    z
>   -- - --
    a     3
         a
In[9]:=
Kauffman[Link[11, NonAlternating, 98]][a, z]
Out[9]=   
                                                                  2    2
  -10   7    7     3      7      4    z    8 z   29 z   20 z   2 z    z
-a    - -- - -- + ---- + ---- + --- + -- - --- - ---- - ---- + ---- + -- + 
         4    2    5      3     a z    7    5      3     a      10     8
        a    a    a  z   a  z         a    a      a            a      a
 
        2       2      3      3       3       3    4       4    4       4
    24 z    23 z    3 z    6 z    46 z    37 z    z     2 z    z    26 z
>   ----- + ----- + ---- + ---- + ----- + ----- - --- + ---- + -- - ----- - 
      4       2       9      5      3       a      10     8     6     4
     a       a       a      a      a              a      a     a     a
 
        4      5      5       5       5      6      6      6      6    7
    24 z    2 z    3 z    33 z    28 z    2 z    2 z    9 z    9 z    z
>   ----- - ---- - ---- - ----- - ----- - ---- - ---- + ---- + ---- - -- + 
      2       9      5      3       a       8      6      4      2     7
     a       a      a      a               a      a      a      a     a
 
        7      7    8    8    9    9
    10 z    9 z    z    z    z    z
>   ----- + ---- - -- - -- - -- - --
      3      a      4    2    3   a
     a             a    a    a
In[10]:=
Kh[L][q, t]
Out[10]=   
                                     4
   2      4     1       1      -2   q       4      6      8        6  2
2 q  + 2 q  + ----- + ----- + t   + -- + 2 q  t + q  t + q  t + 3 q  t  + 
               4  4    2  4         t
              q  t    q  t
 
       8  2    8  3      10  3      10  4    12  4    12  5      14  5
>   2 q  t  + q  t  + 3 q   t  + 3 q   t  + q   t  + q   t  + 3 q   t  + 
 
     14  6    16  6    18  7
>   q   t  + q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n98
L11n97
L11n97
L11n99
L11n99