 L10a108 |
 L10a110 |
| © | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: |
|
 Knotscape |
This page is passe. Go here
instead!
The 2-Component Link L10a109Visit L10a109's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,14,19,13 X20,16,11,15 X14,20,15,19 X2,11,3,12 X8,18,9,17 |
| Gauss Code: | {{1, -9, 4, -5, 3, -4, 2, -10, 5, -3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}} |
| Jones Polynomial: | q-9/2 - 2q-7/2 + 5q-5/2 - 9q-3/2 + 10q-1/2 - 13q1/2 + 11q3/2 - 10q5/2 + 7q7/2 - 3q9/2 + q11/2 |
| A2 (sl(3)) Invariant: | - q-14 - q-12 - q-10 - 3q-8 + 2q-6 + q-4 + 4q-2 + 7 + 2q2 + 5q4 - 2q6 - q10 - 3q12 + q14 - q16 |
| HOMFLY-PT Polynomial: | 2a-3z-1 + 4a-3z + 3a-3z3 + a-3z5 - 7a-1z-1 - 13a-1z - 11a-1z3 - 5a-1z5 - a-1z7 + 7az-1 + 12az + 8az3 + 2az5 - 2a3z-1 - 3a3z - a3z3 |
| Kauffman Polynomial: | a-6z2 - a-6z4 + 2a-5z3 - 3a-5z5 + 2a-4 - 6a-4z2 + 8a-4z4 - 6a-4z6 - 2a-3z-1 + 6a-3z - 10a-3z3 + 11a-3z5 - 7a-3z7 + 8a-2 - 20a-2z2 + 17a-2z4 - 2a-2z6 - 4a-2z8 - 7a-1z-1 + 18a-1z - 27a-1z3 + 27a-1z5 - 10a-1z7 - a-1z9 + 13 - 25z2 + 15z4 + 6z6 - 6z8 - 7az-1 + 16az - 21az3 + 19az5 - 5az7 - az9 + 8a2 - 17a2z2 + 11a2z4 + a2z6 - 2a2z8 - 2a3z-1 + 4a3z - 6a3z3 + 6a3z5 - 2a3z7 + 2a4 - 5a4z2 + 4a4z4 - a4z6 |
| Khovanov Homology: |
|
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, 109]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, Alternating, 109]] |
Out[4]= | PD[X[12, 1, 13, 2], X[16, 8, 17, 7], X[10, 5, 1, 6], X[6, 3, 7, 4], > X[4, 9, 5, 10], X[18, 14, 19, 13], X[20, 16, 11, 15], X[14, 20, 15, 19], > X[2, 11, 3, 12], X[8, 18, 9, 17]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 4, -5, 3, -4, 2, -10, 5, -3},
> {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(9/2) 2 5 9 10 3/2 5/2
q - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 11 q - 10 q +
7/2 5/2 3/2 Sqrt[q]
q q q
7/2 9/2 11/2
> 7 q - 3 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -14 -12 -10 3 2 -4 4 2 4 6 10
7 - q - q - q - -- + -- + q + -- + 2 q + 5 q - 2 q - q -
8 6 2
q q q
12 14 16
> 3 q + q - q |
In[8]:= | HOMFLYPT[Link[10, Alternating, 109]][a, z] |
Out[8]= | 3 3 3
2 7 7 a 2 a 4 z 13 z 3 3 z 11 z
---- - --- + --- - ---- + --- - ---- + 12 a z - 3 a z + ---- - ----- +
3 a z z z 3 a 3 a
a z a a
5 5 7
3 3 3 z 5 z 5 z
> 8 a z - a z + -- - ---- + 2 a z - --
3 a a
a |
In[9]:= | Kauffman[Link[10, Alternating, 109]][a, z] |
Out[9]= | 3
2 8 2 4 2 7 7 a 2 a 6 z 18 z
13 + -- + -- + 8 a + 2 a - ---- - --- - --- - ---- + --- + ---- + 16 a z +
4 2 3 a z z z 3 a
a a a z a
2 2 2 3 3
3 2 z 6 z 20 z 2 2 4 2 2 z 10 z
> 4 a z - 25 z + -- - ---- - ----- - 17 a z - 5 a z + ---- - ----- -
6 4 2 5 3
a a a a a
3 4 4 4
27 z 3 3 3 4 z 8 z 17 z 2 4
> ----- - 21 a z - 6 a z + 15 z - -- + ---- + ----- + 11 a z +
a 6 4 2
a a a
5 5 5 6 6
4 4 3 z 11 z 27 z 5 3 5 6 6 z 2 z
> 4 a z - ---- + ----- + ----- + 19 a z + 6 a z + 6 z - ---- - ---- +
5 3 a 4 2
a a a a
7 7 8
2 6 4 6 7 z 10 z 7 3 7 8 4 z 2 8
> a z - a z - ---- - ----- - 5 a z - 2 a z - 6 z - ---- - 2 a z -
3 a 2
a a
9
z 9
> -- - a z
a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 1 1 1 4 1 5 4 5 5
8 + 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
> 5 q t + 6 q t + 5 q t + 5 q t + 2 q t + 5 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 L10a109 |
|