 L10n100 |
 L10n102 |
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 Knotscape |
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The 4-Component Link L10n101Visit L10n101's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X20,16,17,15 X16,20,9,19 X12,18,13,17 X2536 X9,1,10,4 |
| Gauss Code: | {{1, -9, -2, 10}, {9, -1, -3, 4}, {8, -5, 7, -6}, {-10, 2, 5, -8, -4, 3, 6, -7}} |
| Jones Polynomial: | - q-1/2 + 2q1/2 - 4q3/2 + 2q5/2 - 4q7/2 + 2q9/2 - 3q11/2 - q13/2 - q17/2 |
| A2 (sl(3)) Invariant: | q-2 + q2 + 2q4 + 2q6 + 5q8 + 4q10 + 8q12 + 9q14 + 11q16 + 12q18 + 9q20 + 8q22 + 5q24 + 3q26 + q28 |
| HOMFLY-PT Polynomial: | - a-9z-3 - a-9z-1 + 3a-7z-3 + 5a-7z-1 + 2a-7z - 3a-5z-3 - 6a-5z-1 - 3a-5z + a-3z-3 + a-3z-1 - a-3z - 3a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3 |
| Kauffman Polynomial: | a-9z-3 - 6a-9z-1 + 14a-9z - 15a-9z3 + 7a-9z5 - a-9z7 - 3a-8z-2 + 13a-8 - 19a-8z2 + 9a-8z4 - a-8z6 + 3a-7z-3 - 14a-7z-1 + 27a-7z - 28a-7z3 + 10a-7z5 - a-7z7 - 6a-6z-2 + 24a-6 - 33a-6z2 + 14a-6z4 - 2a-6z6 + 3a-5z-3 - 12a-5z-1 + 16a-5z - 10a-5z3 + 4a-5z5 - a-5z7 - 3a-4z-2 + 11a-4 - 14a-4z2 + 10a-4z4 - 3a-4z6 + a-3z-3 - 3a-3z-1 + 6a-3z3 - a-3z7 - a-2 + 5a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5 |
| 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]= | 4 |
In[3]:= | Show[DrawMorseLink[Link[10, NonAlternating, 101]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, NonAlternating, 101]] |
Out[4]= | PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[7, 15, 8, 14], X[13, 5, 14, 8], > X[18, 12, 19, 11], X[20, 16, 17, 15], X[16, 20, 9, 19], X[12, 18, 13, 17], > X[2, 5, 3, 6], X[9, 1, 10, 4]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, -2, 10}, {9, -1, -3, 4}, {8, -5, 7, -6},
> {-10, 2, 5, -8, -4, 3, 6, -7}] |
In[6]:= | Jones[L][q] |
Out[6]= | 1 3/2 5/2 7/2 9/2 11/2 13/2
-(-------) + 2 Sqrt[q] - 4 q + 2 q - 4 q + 2 q - 3 q - q -
Sqrt[q]
17/2
> q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -2 2 4 6 8 10 12 14 16 18
q + q + 2 q + 2 q + 5 q + 4 q + 8 q + 9 q + 11 q + 12 q +
20 22 24 26 28
> 9 q + 8 q + 5 q + 3 q + q |
In[8]:= | HOMFLYPT[Link[10, NonAlternating, 101]][a, z] |
Out[8]= | 1 3 3 1 1 5 6 1 1 2 z
-(-----) + ----- - ----- + ----- - ---- + ---- - ---- + ---- + --- + --- -
9 3 7 3 5 3 3 3 9 7 5 3 a z 7
a z a z a z a z a z a z a z a z a
3 3 5
3 z z 2 z 3 z z z
> --- - -- + --- - ---- + -- - --
5 3 a 3 a 3
a a a a |
In[9]:= | Kauffman[Link[10, NonAlternating, 101]][a, z] |
Out[9]= | 13 24 11 -2 1 3 3 1 3 6 3
-- + -- + -- - a + ----- + ----- + ----- + ----- - ----- - ----- - ----- -
8 6 4 9 3 7 3 5 3 3 3 8 2 6 2 4 2
a a a a z a z a z a z a z a z a z
2
6 14 12 3 1 14 z 27 z 16 z 3 z 19 z
> ---- - ---- - ---- - ---- + --- + ---- + ---- + ---- - --- - ----- -
9 7 5 3 a z 9 7 5 a 8
a z a z a z a z a a a a
2 2 3 3 3 3 3 4 4
33 z 14 z 15 z 28 z 10 z 6 z 3 z 9 z 14 z
> ----- - ----- - ----- - ----- - ----- + ---- + ---- + ---- + ----- +
6 4 9 7 5 3 a 8 6
a a a a a a a a
4 4 5 5 5 5 6 6 6 6 7
10 z 5 z 7 z 10 z 4 z z z 2 z 3 z 2 z z
> ----- + ---- + ---- + ----- + ---- - -- - -- - ---- - ---- - ---- - -- -
4 2 9 7 5 a 8 6 4 2 9
a a a a a a a a a a
7 7 7
z z z
> -- - -- - --
7 5 3
a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2
2 4 1 1 q 4 6 2 8 2 6 3 10 3
3 q + 4 q + ----- + - + -- + 2 q t + 3 q t + 2 q t + q t + 3 q t +
2 2 t t
q t
10 4 12 4 10 5 14 6 16 8 18 8
> 5 q t + 4 q t + q t + q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n101 |
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