© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L11a64
L11a64
L11a66
L11a66
L11a65
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L11a65

Visit L11a65's page at Knotilus!

Acknowledgement

L11a65 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-1/2 + 3q1/2 - 7q3/2 + 10q5/2 - 15q7/2 + 16q9/2 - 17q11/2 + 14q13/2 - 10q15/2 + 7q17/2 - 3q19/2 + q21/2

A2 (sl(3)) Invariant: q-2 - 1 + 3q4 - 2q6 + 4q8 + 3q10 + q12 + 5q14 + 3q18 - q20 - 3q22 + q24 - 4q26 - q28 + q30 - q32

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

Kauffman Polynomial: - 2a-12z2 + 3a-12z4 - a-12z6 - 4a-11z3 + 8a-11z5 - 3a-11z7 - 4a-10 + 12a-10z2 - 16a-10z4 + 15a-10z6 - 5a-10z8 + a-9z-1 + 2a-9z3 - 4a-9z5 + 8a-9z7 - 4a-9z9 - 9a-8 + 31a-8z2 - 46a-8z4 + 34a-8z6 - 9a-8z8 - a-8z10 + a-7z-1 + 5a-7z - 14a-7z3 + 12a-7z7 - 7a-7z9 - 4a-6 + 15a-6z2 - 32a-6z4 + 27a-6z6 - 9a-6z8 - a-6z10 - 2a-5z-1 + 13a-5z - 29a-5z3 + 22a-5z5 - 4a-5z7 - 3a-5z9 + 2a-4 - 3a-4z2 + 6a-4z6 - 5a-4z8 - 2a-3z-1 + 7a-3z - 7a-3z3 + 9a-3z5 - 5a-3z7 - a-2z2 + 5a-2z4 - 3a-2z6 - a-1z + 2a-1z3 - a-1z5

Khovanov Homology:
trqj r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6r = 7r = 8r = 9
j = 22           1
j = 20          2 
j = 18         51 
j = 16        52  
j = 14       95   
j = 12      85    
j = 10     89     
j = 8    78      
j = 6   38       
j = 4  47        
j = 2 15         
j = 0 2          
j = -21           


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, 65]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[11, Alternating, 65]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[20, 12, 21, 11], 
 
>   X[22, 16, 5, 15], X[16, 22, 17, 21], X[8, 20, 9, 19], X[18, 10, 19, 9], 
 
>   X[10, 18, 11, 17], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 2, -11}, {10, -1, 3, -7, 8, -9, 4, -2, 11, -3, 5, -6, 9, -8, 
 
>    7, -4, 6, -5}]
In[6]:=
Jones[L][q]
Out[6]=   
     1                      3/2       5/2       7/2       9/2       11/2
-(-------) + 3 Sqrt[q] - 7 q    + 10 q    - 15 q    + 16 q    - 17 q     + 
  Sqrt[q]
 
        13/2       15/2      17/2      19/2    21/2
>   14 q     - 10 q     + 7 q     - 3 q     + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
      -2      4      6      8      10    12      14      18    20      22
-1 + q   + 3 q  - 2 q  + 4 q  + 3 q   + q   + 5 q   + 3 q   - q   - 3 q   + 
 
     24      26    28    30    32
>   q   - 4 q   - q   + q   - q
In[8]:=
HOMFLYPT[Link[11, Alternating, 65]][a, z]
Out[8]=   
                                                       3    3      3    3
 1      1      2      2     z    z    4 z   3 z   z   z    z    4 z    z
---- - ---- - ---- + ---- + -- - -- - --- + --- + - + -- - -- - ---- + -- - 
 9      7      5      3      9    7    5     3    a    9    7     5    a
a  z   a  z   a  z   a  z   a    a    a     a         a    a     a
 
     5      5    5
    z    2 z    z
>   -- - ---- - --
     7     5     3
    a     a     a
In[9]:=
Kauffman[Link[11, Alternating, 65]][a, z]
Out[9]=   
                                                                           2
-4    9    4    2     1      1      2      2     5 z   13 z   7 z   z   2 z
--- - -- - -- + -- + ---- + ---- - ---- - ---- + --- + ---- + --- - - - ---- + 
 10    8    6    4    9      7      5      3      7      5     3    a    12
a     a    a    a    a  z   a  z   a  z   a  z   a      a     a         a
 
        2       2       2      2    2      3      3       3       3      3
    12 z    31 z    15 z    3 z    z    4 z    2 z    14 z    29 z    7 z
>   ----- + ----- + ----- - ---- - -- - ---- + ---- - ----- - ----- - ---- + 
      10      8       6       4     2    11      9      7       5       3
     a       a       a       a     a    a       a      a       a       a
 
       3      4       4       4       4      4      5      5       5      5
    2 z    3 z    16 z    46 z    32 z    5 z    8 z    4 z    22 z    9 z
>   ---- + ---- - ----- - ----- - ----- + ---- + ---- - ---- + ----- + ---- - 
     a      12      10      8       6       2     11      9      5       3
           a       a       a       a       a     a       a      a       a
 
     5    6        6       6       6      6      6      7      7       7
    z    z     15 z    34 z    27 z    6 z    3 z    3 z    8 z    12 z
>   -- - --- + ----- + ----- + ----- + ---- - ---- - ---- + ---- + ----- - 
    a     12     10      8       6       4      2     11      9      7
         a      a       a       a       a      a     a       a      a
 
       7      7      8      8      8      8      9      9      9    10    10
    4 z    5 z    5 z    9 z    9 z    5 z    4 z    7 z    3 z    z     z
>   ---- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - ---- - --- - ---
      5      3     10      8      6      4      9      7      5     8     6
     a      a     a       a      a      a      a      a      a     a     a
In[10]:=
Kh[L][q, t]
Out[10]=   
                           2
   2      4     1     2   q       4        6        6  2      8  2      8  3
5 q  + 4 q  + ----- + - + -- + 7 q  t + 3 q  t + 8 q  t  + 7 q  t  + 8 q  t  + 
               2  2   t   t
              q  t
 
       10  3      10  4      12  4      12  5      14  5      14  6
>   8 q   t  + 9 q   t  + 8 q   t  + 5 q   t  + 9 q   t  + 5 q   t  + 
 
       16  6      16  7      18  7    18  8      20  8    22  9
>   5 q   t  + 2 q   t  + 5 q   t  + q   t  + 2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a65
L11a64
L11a64
L11a66
L11a66