© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L10a94
L10a94
L10a96
L10a96
L10a95
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L10a95

Visit L10a95's page at Knotilus!

Acknowledgement

L10a95 as Morse Link
DrawMorseLink

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

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

Jones Polynomial: - q-13/2 + 3q-11/2 - 5q-9/2 + 9q-7/2 - 12q-5/2 + 11q-3/2 - 12q-1/2 + 9q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant: q-20 - q-18 + q-14 - 4q-12 + q-10 + q-8 + q-6 + 4q-4 + 4 + 2q6 - 2q8 + q12

HOMFLY-PT Polynomial: - a-3z - a-1z-1 + a-1z + 2a-1z3 + az-1 + 2az - az5 - 3a3z - 2a3z3 - a3z5 + a5z + a5z3

Kauffman Polynomial: - a-3z + 2a-3z3 - a-3z5 - 3a-2z2 + 6a-2z4 - 3a-2z6 - a-1z-1 + 2a-1z - 2a-1z3 + 6a-1z5 - 4a-1z7 + 1 - 4z2 + 7z4 - 3z8 - az-1 + 10az - 17az3 + 17az5 - 7az7 - az9 - 5a2z2 + a2z4 + 7a2z6 - 6a2z8 + 10a3z - 21a3z3 + 19a3z5 - 7a3z7 - a3z9 - 8a4z2 + 7a4z4 + a4z6 - 3a4z8 + 3a5z - 6a5z3 + 8a5z5 - 4a5z7 - 4a6z2 + 7a6z4 - 3a6z6 + 2a7z3 - a7z5

Khovanov Homology:
trqj r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 8          1
j = 6         2 
j = 4        41 
j = 2       52  
j = 0      74   
j = -2     67    
j = -4    65     
j = -6   36      
j = -8  26       
j = -10 13        
j = -12 2         
j = -141          


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, 95]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, Alternating, 95]]
Out[4]=   
PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 6, 15, 5], X[16, 13, 17, 14], 
 
>   X[18, 8, 19, 7], X[20, 17, 9, 18], X[4, 16, 5, 15], X[6, 20, 7, 19], 
 
>   X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 2, -7, 3, -8, 5, -10}, 
 
>   {9, -1, 10, -2, 4, -3, 7, -4, 6, -5, 8, -6}]
In[6]:=
Jones[L][q]
Out[6]=   
  -(13/2)     3      5      9      12     11      12                     3/2
-q        + ----- - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 6 q    + 
             11/2    9/2    7/2    5/2    3/2   Sqrt[q]
            q       q      q      q      q
 
       5/2    7/2
>   3 q    - q
In[7]:=
A2Invariant[L][q]
Out[7]=   
     -20    -18    -14    4     -10    -8    -6   4       6      8    12
4 + q    - q    + q    - --- + q    + q   + q   + -- + 2 q  - 2 q  + q
                          12                       4
                         q                        q
In[8]:=
HOMFLYPT[Link[10, Alternating, 95]][a, z]
Out[8]=   
                                                 3
   1     a   z    z              3      5     2 z       3  3    5  3      5
-(---) + - - -- + - + 2 a z - 3 a  z + a  z + ---- - 2 a  z  + a  z  - a z  - 
  a z    z    3   a                            a
             a
 
     3  5
>   a  z
In[9]:=
Kauffman[Link[10, Alternating, 95]][a, z]
Out[9]=   
                                                               2
     1    a   z    2 z                3        5        2   3 z       2  2
1 - --- - - - -- + --- + 10 a z + 10 a  z + 3 a  z - 4 z  - ---- - 5 a  z  - 
    a z   z    3    a                                         2
              a                                              a
 
                           3      3
       4  2      6  2   2 z    2 z          3       3  3      5  3      7  3
>   8 a  z  - 4 a  z  + ---- - ---- - 17 a z  - 21 a  z  - 6 a  z  + 2 a  z  + 
                          3     a
                         a
 
              4                                5      5
       4   6 z     2  4      4  4      6  4   z    6 z          5       3  5
>   7 z  + ---- + a  z  + 7 a  z  + 7 a  z  - -- + ---- + 17 a z  + 19 a  z  + 
             2                                 3    a
            a                                 a
 
                         6                                  7
       5  5    7  5   3 z       2  6    4  6      6  6   4 z         7
>   8 a  z  - a  z  - ---- + 7 a  z  + a  z  - 3 a  z  - ---- - 7 a z  - 
                        2                                 a
                       a
 
       3  7      5  7      8      2  8      4  8      9    3  9
>   7 a  z  - 4 a  z  - 3 z  - 6 a  z  - 3 a  z  - a z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    7      1        2        1        3        2       6       3       6
7 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 
     2    14  6    12  5    10  5    10  4    8  4    8  3    6  3    6  2
    q    q   t    q   t    q   t    q   t    q  t    q  t    q  t    q  t
 
      6      5      6              2        2  2      4  2    4  3      6  3
>   ----- + ---- + ---- + 4 t + 5 q  t + 2 q  t  + 4 q  t  + q  t  + 2 q  t  + 
     4  2    4      2
    q  t    q  t   q  t
 
     8  4
>   q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a95
L10a94
L10a94
L10a96
L10a96