© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L10a68
L10a68
L10a70
L10a70
L10a69
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L10a69

Visit L10a69's page at Knotilus!

Acknowledgement

L10a69 as Morse Link
DrawMorseLink

PD Presentation: X8192 X16,9,17,10 X6718 X20,17,7,18 X10,4,11,3 X14,6,15,5 X4,12,5,11 X18,14,19,13 X12,20,13,19 X2,16,3,15

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

Jones Polynomial: - q-7/2 + 4q-5/2 - 8q-3/2 + 11q-1/2 - 15q1/2 + 15q3/2 - 14q5/2 + 11q7/2 - 7q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant: q-10 - 2q-8 + 2q-6 + 5 - 2q2 + 3q4 - 2q6 + 2q10 - 2q12 + 3q14 + q20

HOMFLY-PT Polynomial: - a-5z-1 - 2a-5z - a-5z3 + 2a-3z-1 + 6a-3z + 6a-3z3 + 2a-3z5 - 2a-1z-1 - 4a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + az + 2az3 + az5

Kauffman Polynomial: - a-7z + 2a-7z3 - a-7z5 - 2a-6z2 + 5a-6z4 - 3a-6z6 - a-5z-1 + 3a-5z - 5a-5z3 + 8a-5z5 - 5a-5z7 + 2a-4z2 - 3a-4z4 + 6a-4z6 - 5a-4z8 - 2a-3z-1 + 11a-3z - 23a-3z3 + 23a-3z5 - 8a-3z7 - 2a-3z9 - a-2 + 5a-2z2 - 14a-2z4 + 20a-2z6 - 11a-2z8 - 2a-1z-1 + 9a-1z - 24a-1z3 + 28a-1z5 - 10a-1z7 - 2a-1z9 + 7z6 - 6z8 - az-1 + 2az - 7az3 + 13az5 - 7az7 - a2z2 + 6a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5r = 6
j = 14          1
j = 12         2 
j = 10        51 
j = 8       62  
j = 6      85   
j = 4     87    
j = 2    77     
j = 0   59      
j = -2  36       
j = -4 15        
j = -6 3         
j = -81          


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[10, Alternating, 69]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 69]]
Out[4]=   
PD[X[8, 1, 9, 2], X[16, 9, 17, 10], X[6, 7, 1, 8], X[20, 17, 7, 18], 
 
>   X[10, 4, 11, 3], X[14, 6, 15, 5], X[4, 12, 5, 11], X[18, 14, 19, 13], 
 
>   X[12, 20, 13, 19], X[2, 16, 3, 15]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -10, 5, -7, 6, -3}, 
 
>   {3, -1, 2, -5, 7, -9, 8, -6, 10, -2, 4, -8, 9, -4}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(7/2)    4      8       11                       3/2       5/2       7/2
-q       + ---- - ---- + ------- - 15 Sqrt[q] + 15 q    - 14 q    + 11 q    - 
            5/2    3/2   Sqrt[q]
           q      q
 
       9/2      11/2    13/2
>   7 q    + 3 q     - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -10   2    2       2      4      6      10      12      14    20
5 + q    - -- + -- - 2 q  + 3 q  - 2 q  + 2 q   - 2 q   + 3 q   + q
            8    6
           q    q
In[8]:=
HOMFLYPT[Link[10, Alternating, 69]][a, z]
Out[8]=   
                                                    3      3      3
   1       2      2    a   2 z   6 z   4 z         z    6 z    6 z         3
-(----) + ---- - --- + - - --- + --- - --- + a z - -- + ---- - ---- + 2 a z  + 
   5       3     a z   z    5     3     a           5     3     a
  a  z    a  z             a     a                 a     a
 
       5      5           7
    2 z    4 z       5   z
>   ---- - ---- + a z  - --
      3     a            a
     a
In[9]:=
Kauffman[Link[10, Alternating, 69]][a, z]
Out[9]=   
                                                                  2      2
  -2    1      2      2    a   z    3 z   11 z   9 z           2 z    2 z
-a   - ---- - ---- - --- - - - -- + --- + ---- + --- + 2 a z - ---- + ---- + 
        5      3     a z   z    7    5      3     a              6      4
       a  z   a  z             a    a      a                    a      a
 
       2              3      3       3       3                       4      4
    5 z     2  2   2 z    5 z    23 z    24 z         3    3  3   5 z    3 z
>   ---- - a  z  + ---- - ---- - ----- - ----- - 7 a z  + a  z  + ---- - ---- - 
      2              7      5      3       a                        6      4
     a              a      a      a                                a      a
 
        4              5      5       5       5
    14 z       2  4   z    8 z    23 z    28 z          5    3  5      6
>   ----- + 6 a  z  - -- + ---- + ----- + ----- + 13 a z  - a  z  + 7 z  - 
      2                7     5      3       a
     a                a     a      a
 
       6      6       6                7      7       7
    3 z    6 z    20 z       2  6   5 z    8 z    10 z         7      8
>   ---- + ---- + ----- - 4 a  z  - ---- - ---- - ----- - 7 a z  - 6 z  - 
      6      4      2                 5      3      a
     a      a      a                 a      a
 
       8       8      9      9
    5 z    11 z    2 z    2 z
>   ---- - ----- - ---- - ----
      4      2       3     a
     a      a       a
In[10]:=
Kh[L][q, t]
Out[10]=   
       2     1       3       1       5       3     5    6        2        4
9 + 7 q  + ----- + ----- + ----- + ----- + ----- + - + ---- + 7 q  t + 8 q  t + 
            8  4    6  3    4  3    4  2    2  2   t    2
           q  t    q  t    q  t    q  t    q  t        q  t
 
       4  2      6  2      6  3      8  3      8  4      10  4    10  5
>   7 q  t  + 8 q  t  + 5 q  t  + 6 q  t  + 2 q  t  + 5 q   t  + q   t  + 
 
       12  5    14  6
>   2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a69
L10a68
L10a68
L10a70
L10a70