 L11n255 |
 L11n257 |
| © | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: |
|
 Knotscape |
This page is passe. Go here
instead!
The 3-Component Link L11n256Visit L11n256's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X3,11,4,10 X16,12,17,11 X18,22,19,21 X20,14,21,13 X12,20,13,19 X22,18,9,17 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4 |
| Gauss Code: | {{1, -10, -2, 11}, {10, -1, -9, 8}, {-11, 2, 3, -6, 5, 9, -8, -3, 7, -4, 6, -5, 4, -7}} |
| Jones Polynomial: | q-3 - q-2 + 3q-1 - 2 + 4q - q2 + 2q3 - q4 - q5 + q6 - q7 |
| A2 (sl(3)) Invariant: | q-10 + q-8 + 2q-6 + 4q-4 + 4q-2 + 6 + 6q2 + 6q4 + 6q6 + 2q8 + q10 - 3q12 - 3q14 - 2q16 - 2q18 - q20 - q22 |
| HOMFLY-PT Polynomial: | - a-6z-2 - 2a-6 - a-6z2 + 3a-4z-2 + 6a-4 + 5a-4z2 + a-4z4 - 2a-2z-2 - 3a-2 - 3a-2z2 - a-2z4 - z-2 - 3 - 3z2 - z4 + a2z-2 + 2a2 + a2z2 |
| Kauffman Polynomial: | 2a-7z-1 - 7a-7z + 10a-7z3 - 6a-7z5 + a-7z7 - a-6z-2 + a-6 - 2a-6z2 + 8a-6z4 - 6a-6z6 + a-6z8 + 8a-5z-1 - 27a-5z + 35a-5z3 - 16a-5z5 + 2a-5z7 - 3a-4z-2 + 5a-4 - 7a-4z2 + 12a-4z4 - 7a-4z6 + a-4z8 + 10a-3z-1 - 34a-3z + 44a-3z3 - 21a-3z5 + 3a-3z7 - 2a-2z-2 + 4a-2 - 8a-2z2 + 8a-2z4 - 5a-2z6 + a-2z8 + 2a-1z-1 - 10a-1z + 18a-1z3 - 14a-1z5 + 3a-1z7 + z-2 - 3 + 4z2 - z4 - 3z6 + z8 - 2az-1 + 4az - az3 - 3az5 + az7 + a2z-2 - 4a2 + 7a2z2 - 5a2z4 + a2z6 |
| 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]= | 3 |
In[3]:= | Show[DrawMorseLink[Link[11, NonAlternating, 256]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 256]] |
Out[4]= | PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[16, 12, 17, 11], X[18, 22, 19, 21], > X[20, 14, 21, 13], X[12, 20, 13, 19], X[22, 18, 9, 17], X[15, 8, 16, 5], > X[7, 14, 8, 15], X[2, 5, 3, 6], X[9, 1, 10, 4]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, -2, 11}, {10, -1, -9, 8},
> {-11, 2, 3, -6, 5, 9, -8, -3, 7, -4, 6, -5, 4, -7}] |
In[6]:= | Jones[L][q] |
Out[6]= | -3 -2 3 2 3 4 5 6 7
-2 + q - q + - + 4 q - q + 2 q - q - q + q - q
q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -10 -8 2 4 4 2 4 6 8 10 12
6 + q + q + -- + -- + -- + 6 q + 6 q + 6 q + 2 q + q - 3 q -
6 4 2
q q q
14 16 18 20 22
> 3 q - 2 q - 2 q - q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 256]][a, z] |
Out[8]= | 2 2
2 6 3 2 -2 1 3 2 a 2 z
-3 - -- + -- - -- + 2 a - z - ----- + ----- - ----- + -- - 3 z - -- +
6 4 2 6 2 4 2 2 2 2 6
a a a a z a z a z z a
2 2 4 4
5 z 3 z 2 2 4 z z
> ---- - ---- + a z - z + -- - --
4 2 4 2
a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 256]][a, z] |
Out[9]= | 2
-6 5 4 2 -2 1 3 2 a 2 8
-3 + a + -- + -- - 4 a + z - ----- - ----- - ----- + -- + ---- + ---- +
4 2 6 2 4 2 2 2 2 7 5
a a a z a z a z z a z a z
2 2
10 2 2 a 7 z 27 z 34 z 10 z 2 2 z 7 z
> ---- + --- - --- - --- - ---- - ---- - ---- + 4 a z + 4 z - ---- - ---- -
3 a z z 7 5 3 a 6 4
a z a a a a a
2 3 3 3 3 4 4
8 z 2 2 10 z 35 z 44 z 18 z 3 4 8 z 12 z
> ---- + 7 a z + ----- + ----- + ----- + ----- - a z - z + ---- + ----- +
2 7 5 3 a 6 4
a a a a a a
4 5 5 5 5 6
8 z 2 4 6 z 16 z 21 z 14 z 5 6 6 z
> ---- - 5 a z - ---- - ----- - ----- - ----- - 3 a z - 3 z - ---- -
2 7 5 3 a 6
a a a a a
6 6 7 7 7 7 8 8 8
7 z 5 z 2 6 z 2 z 3 z 3 z 7 8 z z z
> ---- - ---- + a z + -- + ---- + ---- + ---- + a z + z + -- + -- + --
4 2 7 5 3 a 6 4 2
a a a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 1 3 1 1 1 2 1 1 2 q
- + 5 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + q t +
q 7 4 5 4 5 3 3 2 2 q t t
q t q t q t q t q t
3 5 3 2 5 2 7 2 5 3 7 3 9 3 7 4
> q t + 2 q t + q t + 4 q t + q t + q t + q t + 3 q t + q t +
9 4 11 5 11 6 15 7
> q t + q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n256 |
|