© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:
L10n37
L10n37
L10n39
L10n39
L10n38
Knotscape
This page is passe. Go here instead!

The 2-Component Link

L10n38

Visit L10n38's page at Knotilus!

Acknowledgement

L10n38 as Morse Link
DrawMorseLink

PD Presentation: X6172 X12,3,13,4 X16,13,17,14 X14,9,15,10 X10,15,11,16 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,11,1,12

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

Jones Polynomial: q-15/2 - 2q-13/2 + 4q-11/2 - 6q-9/2 + 5q-7/2 - 7q-5/2 + 5q-3/2 - 4q-1/2 + 2q1/2

A2 (sl(3)) Invariant: - q-24 - 2q-22 - q-20 - 2q-18 + 2q-16 + 4q-14 + 4q-12 + 6q-10 + 2q-8 + 2q-6 - q-4 - 2q-2 - 2q2

HOMFLY-PT Polynomial: 2az-1 + 4az + 2az3 - 7a3z-1 - 14a3z - 9a3z3 - 2a3z5 + 7a5z-1 + 9a5z + 3a5z3 - 2a7z-1 - a7z

Kauffman Polynomial: 2 - 3z2 - 2az-1 + 3az - 3az3 - az5 + 8a2 - 17a2z2 + 11a2z4 - 4a2z6 - 7a3z-1 + 16a3z - 18a3z3 + 12a3z5 - 4a3z7 + 13a4 - 29a4z2 + 24a4z4 - 5a4z6 - a4z8 - 7a5z-1 + 17a5z - 22a5z3 + 20a5z5 - 6a5z7 + 8a6 - 20a6z2 + 17a6z4 - 2a6z6 - a6z8 - 2a7z-1 + 4a7z - 7a7z3 + 7a7z5 - 2a7z7 + 2a8 - 5a8z2 + 4a8z4 - a8z6

Khovanov Homology:
trqj r = -7r = -6r = -5r = -4r = -3r = -2r = -1r = 0r = 1
j = 2        2
j = 0       2 
j = -2      43 
j = -4     31  
j = -6    24   
j = -8   43    
j = -10  13     
j = -12 13      
j = -14 1       
j = -161        


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, NonAlternating, 38]]]
Out[3]=   
 -Graphics- 
In[4]:=
PD[L = Link[10, NonAlternating, 38]]
Out[4]=   
PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 13, 17, 14], X[14, 9, 15, 10], 
 
>   X[10, 15, 11, 16], X[17, 5, 18, 20], X[7, 19, 8, 18], X[19, 9, 20, 8], 
 
>   X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=
GaussCode[L]
Out[5]=   
GaussCode[{1, -9, 2, -10}, {9, -1, -7, 8, 4, -5, 10, -2, 3, -4, 5, -3, -6, 7, 
 
>    -8, 6}]
In[6]:=
Jones[L][q]
Out[6]=   
 -(15/2)     2       4      6      5      7      5        4
q        - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q]
            13/2    11/2    9/2    7/2    5/2    3/2   Sqrt[q]
           q       q       q      q      q      q
In[7]:=
A2Invariant[L][q]
Out[7]=   
  -24    2     -20    2     2     4     4     6    2    2     -4   2       2
-q    - --- - q    - --- + --- + --- + --- + --- + -- + -- - q   - -- - 2 q
         22           18    16    14    12    10    8    6          2
        q            q     q     q     q     q     q    q          q
In[8]:=
HOMFLYPT[Link[10, NonAlternating, 38]][a, z]
Out[8]=   
         3      5      7
2 a   7 a    7 a    2 a                3        5      7          3      3  3
--- - ---- + ---- - ---- + 4 a z - 14 a  z + 9 a  z - a  z + 2 a z  - 9 a  z  + 
 z     z      z      z
 
       5  3      3  5
>   3 a  z  - 2 a  z
In[9]:=
Kauffman[Link[10, NonAlternating, 38]][a, z]
Out[9]=   
                                          3      5      7
       2       4      6      8   2 a   7 a    7 a    2 a                3
2 + 8 a  + 13 a  + 8 a  + 2 a  - --- - ---- - ---- - ---- + 3 a z + 16 a  z + 
                                  z     z      z      z
 
        5        7        2       2  2       4  2       6  2      8  2
>   17 a  z + 4 a  z - 3 z  - 17 a  z  - 29 a  z  - 20 a  z  - 5 a  z  - 
 
         3       3  3       5  3      7  3       2  4       4  4       6  4
>   3 a z  - 18 a  z  - 22 a  z  - 7 a  z  + 11 a  z  + 24 a  z  + 17 a  z  + 
 
       8  4      5       3  5       5  5      7  5      2  6      4  6
>   4 a  z  - a z  + 12 a  z  + 20 a  z  + 7 a  z  - 4 a  z  - 5 a  z  - 
 
       6  6    8  6      3  7      5  7      7  7    4  8    6  8
>   2 a  z  - a  z  - 4 a  z  - 6 a  z  - 2 a  z  - a  z  - a  z
In[10]:=
Kh[L][q, t]
Out[10]=   
    3      1        1        1        3        1        3        4       3
2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 
     2    16  7    14  6    12  6    12  5    10  5    10  4    8  4    8  3
    q    q   t    q   t    q   t    q   t    q   t    q   t    q  t    q  t
 
      2       4       3      1      4        2
>   ----- + ----- + ----- + ---- + ---- + 2 q  t
     6  3    6  2    4  2    4      2
    q  t    q  t    q  t    q  t   q  t


Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n38
L10n37
L10n37
L10n39
L10n39