© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L10a102
L10a102
L10a104
L10a104
L10a103
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L10a103

Visit L10a103's page at Knotilus!

Acknowledgement

L10a103 as Morse Link
DrawMorseLink

PD Presentation: X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X4,17,5,18 X18,5,19,6 X6,14,7,13 X14,8,15,7 X8,16,1,15 X16,19,17,20

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

Jones Polynomial: q-9/2 - 3q-7/2 + 5q-5/2 - 9q-3/2 + 11q-1/2 - 12q1/2 + 11q3/2 - 10q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant: - q-14 + q-10 + 4q-6 + 1 - 2q2 + 4q4 + 3q8 + q10 - 2q12 + q14 - q16

HOMFLY-PT Polynomial: 3a-3z + 3a-3z3 + a-3z5 - a-1z-1 - 8a-1z - 10a-1z3 - 5a-1z5 - a-1z7 + az-1 + 7az + 7az3 + 2az5 - 2a3z - a3z3

Kauffman Polynomial: a-6z2 - a-6z4 - a-5z + 3a-5z3 - 3a-5z5 - 2a-4z2 + 5a-4z4 - 5a-4z6 + 2a-3z - 6a-3z3 + 8a-3z5 - 6a-3z7 - 3a-2z2 + 5a-2z4 + a-2z6 - 4a-2z8 - a-1z-1 + 10a-1z - 26a-1z3 + 28a-1z5 - 10a-1z7 - a-1z9 + 1 - 4z4 + 13z6 - 7z8 - az-1 + 10az - 27az3 + 27az5 - 7az7 - az9 - 2a2z2 + 6a2z6 - 3a2z8 + 3a3z - 10a3z3 + 10a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - a4z6

Khovanov Homology:
trqj r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4r = 5
j = 12          1
j = 10         2 
j = 8        41 
j = 6       62  
j = 4      54   
j = 2     76    
j = 0    67     
j = -2   35      
j = -4  26       
j = -6 13        
j = -8 2         
j = -101          


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, 103]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 103]]
Out[4]=   
PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 12, 9, 11], X[2, 9, 3, 10], 
 
>   X[4, 17, 5, 18], X[18, 5, 19, 6], X[6, 14, 7, 13], X[14, 8, 15, 7], 
 
>   X[8, 16, 1, 15], X[16, 19, 17, 20]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -4, 2, -5, 6, -7, 8, -9}, 
 
>   {4, -1, 3, -2, 7, -8, 9, -10, 5, -6, 10, -3}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(9/2)    3      5      9       11                       3/2       5/2
q       - ---- + ---- - ---- + ------- - 12 Sqrt[q] + 11 q    - 10 q    + 
           7/2    5/2    3/2   Sqrt[q]
          q      q      q
 
       7/2      9/2    11/2
>   6 q    - 3 q    + q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -14    -10   4       2      4      8    10      12    14    16
1 - q    + q    + -- - 2 q  + 4 q  + 3 q  + q   - 2 q   + q   - q
                   6
                  q
In[8]:=
HOMFLYPT[Link[10, Alternating, 103]][a, z]
Out[8]=   
                                             3       3                     5
   1     a   3 z   8 z              3     3 z    10 z         3    3  3   z
-(---) + - + --- - --- + 7 a z - 2 a  z + ---- - ----- + 7 a z  - a  z  + -- - 
  a z    z    3     a                       3      a                       3
             a                             a                              a
 
       5             7
    5 z         5   z
>   ---- + 2 a z  - --
     a              a
In[9]:=
Kauffman[Link[10, Alternating, 103]][a, z]
Out[9]=   
                                                   2      2      2
     1    a   z    2 z   10 z               3     z    2 z    3 z       2  2
1 - --- - - - -- + --- + ---- + 10 a z + 3 a  z + -- - ---- - ---- - 2 a  z  - 
    a z   z    5    3     a                        6     4      2
              a    a                              a     a      a
 
                 3      3       3                                4      4
       4  2   3 z    6 z    26 z          3       3  3      4   z    5 z
>   2 a  z  + ---- - ---- - ----- - 27 a z  - 10 a  z  - 4 z  - -- + ---- + 
                5      3      a                                  6     4
               a      a                                         a     a
 
       4                5      5       5                                   6
    5 z       4  4   3 z    8 z    28 z          5       3  5       6   5 z
>   ---- + 3 a  z  - ---- + ---- + ----- + 27 a z  + 10 a  z  + 13 z  - ---- + 
      2                5      3      a                                    4
     a                a      a                                           a
 
     6                        7       7                                8
    z       2  6    4  6   6 z    10 z         7      3  7      8   4 z
>   -- + 6 a  z  - a  z  - ---- - ----- - 7 a z  - 3 a  z  - 7 z  - ---- - 
     2                       3      a                                 2
    a                       a                                        a
 
               9
       2  8   z       9
>   3 a  z  - -- - a z
              a
In[10]:=
Kh[L][q, t]
Out[10]=   
       2     1        2       1       3       2       6       3     6    5
7 + 7 q  + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 
            10  5    8  4    6  4    6  3    4  3    4  2    2  2   t    2
           q   t    q  t    q  t    q  t    q  t    q  t    q  t        q  t
 
       2        4        4  2      6  2      6  3      8  3    8  4
>   6 q  t + 5 q  t + 4 q  t  + 6 q  t  + 2 q  t  + 4 q  t  + q  t  + 
 
       10  4    12  5
>   2 q   t  + q   t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a103
L10a102
L10a102
L10a104
L10a104