 L11n432 |
 L11n434 |
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The 3-Component Link L11n433Visit L11n433's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X5,17,6,16 X11,19,12,18 X17,11,18,10 X19,5,20,4 |
| Gauss Code: | {{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, 8, -10, 9, -11, -7, 6, -5}} |
| Jones Polynomial: | q - q2 + 2q3 + q5 + q6 - q7 + q8 - q9 + q10 |
| A2 (sl(3)) Invariant: | q4 + q6 + q8 + 3q10 + 3q12 + 3q14 + 4q16 + 4q18 + 2q20 + 2q22 + q26 + q30 + 2q32 - q34 |
| HOMFLY-PT Polynomial: | - a-10z2 + a-8z-2 + 3a-8 + 8a-8z2 + 6a-8z4 + a-8z6 - 2a-6z-2 - 9a-6 - 17a-6z2 - 16a-6z4 - 7a-6z6 - a-6z8 + a-4z-2 + 6a-4 + 10a-4z2 + 6a-4z4 + a-4z6 |
| Kauffman Polynomial: | - 3a-12z2 + a-12z4 - 3a-11z3 + a-11z5 - 2a-10 + 5a-10z2 - 5a-10z4 + a-10z6 + 7a-9z3 - 6a-9z5 + a-9z7 - a-8z-2 + 3a-8 - 11a-8z2 + 22a-8z4 - 13a-8z6 + 2a-8z8 + 2a-7z-1 - 9a-7z + 13a-7z3 + a-7z5 - 5a-7z7 + a-7z9 - 2a-6z-2 + 11a-6 - 35a-6z2 + 44a-6z4 - 21a-6z6 + 3a-6z8 + 2a-5z-1 - 9a-5z + 3a-5z3 + 8a-5z5 - 6a-5z7 + a-5z9 - a-4z-2 + 7a-4 - 16a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8 |
| Khovanov Homology: |
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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, 433]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 433]] |
Out[4]= | PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[12, 14, 7, 13], X[2, 7, 3, 8], > X[22, 10, 13, 9], X[6, 22, 1, 21], X[20, 16, 21, 15], X[5, 17, 6, 16], > X[11, 19, 12, 18], X[17, 11, 18, 10], X[19, 5, 20, 4]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3},
> {3, -2, 7, 8, -10, 9, -11, -7, 6, -5}] |
In[6]:= | Jones[L][q] |
Out[6]= | 2 3 5 6 7 8 9 10 q - q + 2 q + q + q - q + q - q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 4 6 8 10 12 14 16 18 20 22 26
q + q + q + 3 q + 3 q + 3 q + 4 q + 4 q + 2 q + 2 q + q +
30 32 34
> q + 2 q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 433]][a, z] |
Out[8]= | 2 2 2 2 4
3 9 6 1 2 1 z 8 z 17 z 10 z 6 z
-- - -- + -- + ----- - ----- + ----- - --- + ---- - ----- + ----- + ---- -
8 6 4 8 2 6 2 4 2 10 8 6 4 8
a a a a z a z a z a a a a a
4 4 6 6 6 8
16 z 6 z z 7 z z z
> ----- + ---- + -- - ---- + -- - --
6 4 8 6 4 6
a a a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 433]][a, z] |
Out[9]= | 2
-2 3 11 7 1 2 1 2 2 9 z 9 z 3 z
--- + -- + -- + -- - ----- - ----- - ----- + ---- + ---- - --- - --- - ---- +
10 8 6 4 8 2 6 2 4 2 7 5 7 5 12
a a a a a z a z a z a z a z a a a
2 2 2 2 3 3 3 3 4 4
5 z 11 z 35 z 16 z 3 z 7 z 13 z 3 z z 5 z
> ---- - ----- - ----- - ----- - ---- + ---- + ----- + ---- + --- - ---- +
10 8 6 4 11 9 7 5 12 10
a a a a a a a a a a
4 4 4 5 5 5 5 6 6 6
22 z 44 z 16 z z 6 z z 8 z z 13 z 21 z
> ----- + ----- + ----- + --- - ---- + -- + ---- + --- - ----- - ----- -
8 6 4 11 9 7 5 10 8 6
a a a a a a a a a a
6 7 7 7 8 8 8 9 9
7 z z 5 z 6 z 2 z 3 z z z z
> ---- + -- - ---- - ---- + ---- + ---- + -- + -- + --
4 9 7 5 8 6 4 7 5
a a a a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 5
5 7 q q 7 9 7 2 9 2 11 2 9 3
2 q + q + -- + -- + q t + q t + 2 q t + 3 q t + q t + q t +
2 t
t
11 3 13 3 11 4 13 4 15 4 13 5 15 5
> q t + q t + 2 q t + 2 q t + q t + q t + 2 q t +
17 5 15 6 17 6 19 7 19 8 21 8
> q t + q t + q t + q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n433 |
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