 L11n106 |
 L11n108 |
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The 2-Component Link L11n107Visit L11n107's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X3,13,4,12 X13,19,14,18 X17,11,18,10 X21,9,22,8 X7,17,8,16 X9,21,10,20 X15,5,16,22 X19,15,20,14 X2536 X11,1,12,4 |
| Gauss Code: | {{1, -10, -2, 11}, {10, -1, -6, 5, -7, 4, -11, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8}} |
| Jones Polynomial: | - 3q3/2 + 6q5/2 - 11q7/2 + 12q9/2 - 14q11/2 + 13q13/2 - 10q15/2 + 7q17/2 - 3q19/2 + q21/2 |
| A2 (sl(3)) Invariant: | 3q4 - q6 + 3q8 + 4q10 + 5q14 + 2q18 - 3q22 + q24 - 4q26 - q28 + q30 - q32 |
| HOMFLY-PT Polynomial: | a-9z-1 + a-9z + a-9z3 - a-7z-1 - a-7z - a-7z3 - a-7z5 - 2a-5z-1 - 5a-5z - 5a-5z3 - 2a-5z5 + 2a-3z-1 + 5a-3z + 3a-3z3 |
| Kauffman Polynomial: | a-12 - 3a-12z2 + 3a-12z4 - a-12z6 + a-11z - 6a-11z3 + 8a-11z5 - 3a-11z7 - 2a-10z2 - a-10z4 + 8a-10z6 - 4a-10z8 + a-9z-1 + 2a-9z - 12a-9z3 + 19a-9z5 - 4a-9z7 - 2a-9z9 - 3a-8 + 6a-8z2 - 11a-8z4 + 21a-8z6 - 10a-8z8 + a-7z-1 + 4a-7z - 19a-7z3 + 25a-7z5 - 8a-7z7 - 2a-7z9 + 2a-6z2 - 7a-6z4 + 9a-6z6 - 6a-6z8 - 2a-5z-1 + 10a-5z - 19a-5z3 + 14a-5z5 - 7a-5z7 + 3a-4 - 3a-4z2 - 3a-4z6 - 2a-3z-1 + 7a-3z - 6a-3z3 |
| 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]= | 2 |
In[3]:= | Show[DrawMorseLink[Link[11, NonAlternating, 107]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 107]] |
Out[4]= | PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[13, 19, 14, 18], X[17, 11, 18, 10], > X[21, 9, 22, 8], X[7, 17, 8, 16], X[9, 21, 10, 20], X[15, 5, 16, 22], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[11, 1, 12, 4]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, -2, 11}, {10, -1, -6, 5, -7, 4, -11, 2, -3, 9, -8, 6, -4, 3,
> -9, 7, -5, 8}] |
In[6]:= | Jones[L][q] |
Out[6]= | 3/2 5/2 7/2 9/2 11/2 13/2 15/2
-3 q + 6 q - 11 q + 12 q - 14 q + 13 q - 10 q +
17/2 19/2 21/2
> 7 q - 3 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 4 6 8 10 14 18 22 24 26 28 30 32 3 q - q + 3 q + 4 q + 5 q + 2 q - 3 q + q - 4 q - q + q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 107]][a, z] |
Out[8]= | 3 3 3 3 5
1 1 2 2 z z 5 z 5 z z z 5 z 3 z z
---- - ---- - ---- + ---- + -- - -- - --- + --- + -- - -- - ---- + ---- - -- -
9 7 5 3 9 7 5 3 9 7 5 3 7
a z a z a z a z a a a a a a a a a
5
2 z
> ----
5
a |
In[9]:= | Kauffman[Link[11, NonAlternating, 107]][a, z] |
Out[9]= | -12 3 3 1 1 2 2 z 2 z 4 z 10 z 7 z
a - -- + -- + ---- + ---- - ---- - ---- + --- + --- + --- + ---- + --- -
8 4 9 7 5 3 11 9 7 5 3
a a a z a z a z a z a a a a a
2 2 2 2 2 3 3 3 3 3
3 z 2 z 6 z 2 z 3 z 6 z 12 z 19 z 19 z 6 z
> ---- - ---- + ---- + ---- - ---- - ---- - ----- - ----- - ----- - ---- +
12 10 8 6 4 11 9 7 5 3
a a a a a a a a a a
4 4 4 4 5 5 5 5 6 6
3 z z 11 z 7 z 8 z 19 z 25 z 14 z z 8 z
> ---- - --- - ----- - ---- + ---- + ----- + ----- + ----- - --- + ---- +
12 10 8 6 11 9 7 5 12 10
a a a a a a a a a a
6 6 6 7 7 7 7 8 8 8
21 z 9 z 3 z 3 z 4 z 8 z 7 z 4 z 10 z 6 z
> ----- + ---- - ---- - ---- - ---- - ---- - ---- - ---- - ----- - ---- -
8 6 4 11 9 7 5 10 8 6
a a a a a a a a a a
9 9
2 z 2 z
> ---- - ----
9 7
a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 4 4 6 6 2 8 2 8 3 10 3
3 q + 2 q + 5 q t + q t + 6 q t + 5 q t + 6 q t + 6 q t +
10 4 12 4 12 5 14 5 14 6 16 6
> 8 q t + 6 q t + 5 q t + 8 q t + 5 q t + 5 q t +
16 7 18 7 18 8 20 8 22 9
> 2 q t + 5 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n107 |
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