 L10n95 |
 L10n97 |
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The 4-Component Link L10n96Visit L10n96's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X2536 X13,15,14,20 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X11,19,12,18 X19,13,20,12 X17,9,18,14 |
| Gauss Code: | {{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -8, 9, -3, 10}, {-7, 6, -10, 8, -9, 3}} |
| Jones Polynomial: | - 3q3/2 + 3q5/2 - 7q7/2 + 6q9/2 - 8q11/2 + 5q13/2 - 5q15/2 + 2q17/2 - q19/2 |
| A2 (sl(3)) Invariant: | 3q4 + 2q6 + 6q8 + 7q10 + 7q12 + 11q14 + 9q16 + 11q18 + 8q20 + 6q22 + 6q24 + 2q26 + 2q28 + q30 |
| HOMFLY-PT Polynomial: | - a-9z-3 - 2a-9z-1 - a-9z + 3a-7z-3 + 9a-7z-1 + 9a-7z + 3a-7z3 - 3a-5z-3 - 12a-5z-1 - 16a-5z - 9a-5z3 - 2a-5z5 + a-3z-3 + 5a-3z-1 + 8a-3z + 3a-3z3 |
| Kauffman Polynomial: | a-11z-1 - 3a-11z + 3a-11z3 - a-11z5 - a-10 + 4a-10z4 - 2a-10z6 + a-9z-3 - 3a-9z-1 + 3a-9z + 2a-9z3 + 2a-9z5 - 2a-9z7 - 3a-8z-2 + 11a-8 - 16a-8z2 + 13a-8z4 - 3a-8z6 - a-8z8 + 3a-7z-3 - 12a-7z-1 + 21a-7z - 26a-7z3 + 18a-7z5 - 6a-7z7 - 6a-6z-2 + 24a-6 - 33a-6z2 + 18a-6z4 - 4a-6z6 - a-6z8 + 3a-5z-3 - 14a-5z-1 + 28a-5z - 31a-5z3 + 15a-5z5 - 4a-5z7 - 3a-4z-2 + 13a-4 - 17a-4z2 + 9a-4z4 - 3a-4z6 + a-3z-3 - 6a-3z-1 + 13a-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]= | 4 |
In[3]:= | Show[DrawMorseLink[Link[10, NonAlternating, 96]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, NonAlternating, 96]] |
Out[4]= | PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[13, 15, 14, 20], X[3, 11, 4, 10], > X[9, 1, 10, 4], X[7, 17, 8, 16], X[15, 5, 16, 8], X[11, 19, 12, 18], > X[19, 13, 20, 12], X[17, 9, 18, 14]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -8, 9, -3, 10},
> {-7, 6, -10, 8, -9, 3}] |
In[6]:= | Jones[L][q] |
Out[6]= | 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2
-3 q + 3 q - 7 q + 6 q - 8 q + 5 q - 5 q + 2 q -
19/2
> q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 4 6 8 10 12 14 16 18 20 22
3 q + 2 q + 6 q + 7 q + 7 q + 11 q + 9 q + 11 q + 8 q + 6 q +
24 26 28 30
> 6 q + 2 q + 2 q + q |
In[8]:= | HOMFLYPT[Link[10, NonAlternating, 96]][a, z] |
Out[8]= | 1 3 3 1 2 9 12 5 z 9 z
-(-----) + ----- - ----- + ----- - ---- + ---- - ---- + ---- - -- + --- -
9 3 7 3 5 3 3 3 9 7 5 3 9 7
a z a z a z a z a z a z a z a z a a
3 3 3 5
16 z 8 z 3 z 9 z 3 z 2 z
> ---- + --- + ---- - ---- + ---- - ----
5 3 7 5 3 5
a a a a a a |
In[9]:= | Kauffman[Link[10, NonAlternating, 96]][a, z] |
Out[9]= | -10 11 24 13 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
1 3 12 14 6 3 z 3 z 21 z 28 z 13 z
> ----- - ---- - ---- - ---- - ---- - --- + --- + ---- + ---- + ---- -
11 9 7 5 3 11 9 7 5 3
a z a z a z a z a z a a a a a
2 2 2 3 3 3 3 3 4 4
16 z 33 z 17 z 3 z 2 z 26 z 31 z 6 z 4 z 13 z
> ----- - ----- - ----- + ---- + ---- - ----- - ----- - ---- + ---- + ----- +
8 6 4 11 9 7 5 3 10 8
a a a a a a a a a a
4 4 5 5 5 5 6 6 6 6
18 z 9 z z 2 z 18 z 15 z 2 z 3 z 4 z 3 z
> ----- + ---- - --- + ---- + ----- + ----- - ---- - ---- - ---- - ---- -
6 4 11 9 7 5 10 8 6 4
a a a a a a a a a a
7 7 7 8 8
2 z 6 z 4 z z z
> ---- - ---- - ---- - -- - --
9 7 5 8 6
a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 4 4 6 2 8 2 8 3 10 3 10 4
3 q + 3 q + 3 q t + 4 q t + 4 q t + 3 q t + 3 q t + 5 q t +
12 4 12 5 14 5 14 6 16 6 16 7 18 7
> 4 q t + q t + 4 q t + 4 q t + 4 q t + q t + q t +
20 8
> q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10n96 |
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