 L11a507 |
 L11a509 |
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 Knotscape |
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The 3-Component Link L11a508Visit L11a508's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X4,21,5,22 X14,6,15,5 X22,15,13,16 X6,14,1,13 |
| Gauss Code: | {{1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7}, {11, -9, 10, -5, 7, -2, 4, -6, 8, -10}} |
| Jones Polynomial: | q-4 - 4q-3 + 8q-2 - 13q-1 + 18 - 19q + 20q2 - 16q3 + 13q4 - 7q5 + 4q6 - q7 |
| A2 (sl(3)) Invariant: | q-12 - 2q-10 + q-8 + q-6 - 4q-4 + 4q-2 - 1 + q2 + 3q4 + 7q8 + 2q10 + 5q12 + 6q14 + 3q18 + q20 - q22 |
| HOMFLY-PT Polynomial: | a-6z-2 - a-6z2 - 2a-4z-2 - 2a-4 + 2a-4z2 + 2a-4z4 + a-2z-2 + 2a-2 + a-2z2 - a-2z4 - a-2z6 - 2z2 - 2z4 - z6 + a2z2 + a2z4 |
| Kauffman Polynomial: | 2a-7z3 - 3a-7z5 + a-7z7 + a-6z-2 - 2a-6 - 5a-6z2 + 17a-6z4 - 15a-6z6 + 4a-6z8 - 2a-5z-1 + 2a-5z + 10a-5z5 - 15a-5z7 + 5a-5z9 + 2a-4z-2 - 3a-4 - 14a-4z2 + 47a-4z4 - 41a-4z6 + 7a-4z8 + 2a-4z10 - 2a-3z-1 + 2a-3z + 2a-3z3 + 14a-3z5 - 30a-3z7 + 12a-3z9 + a-2z-2 - 2a-2 - 11a-2z2 + 42a-2z4 - 49a-2z6 + 14a-2z8 + 2a-2z10 + 10a-1z3 - 16a-1z5 - 3a-1z7 + 7a-1z9 + 4z4 - 15z6 + 11z8 + 4az3 - 13az5 + 11az7 + 2a2z2 - 7a2z4 + 8a2z6 - 2a3z3 + 4a3z5 + a4z4 |
| 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, Alternating, 508]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 508]] |
Out[4]= | PD[X[8, 1, 9, 2], X[18, 12, 19, 11], X[10, 4, 11, 3], X[2, 20, 3, 19], > X[16, 8, 17, 7], X[20, 9, 21, 10], X[12, 18, 7, 17], X[4, 21, 5, 22], > X[14, 6, 15, 5], X[22, 15, 13, 16], X[6, 14, 1, 13]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7},
> {11, -9, 10, -5, 7, -2, 4, -6, 8, -10}] |
In[6]:= | Jones[L][q] |
Out[6]= | -4 4 8 13 2 3 4 5 6 7
18 + q - -- + -- - -- - 19 q + 20 q - 16 q + 13 q - 7 q + 4 q - q
3 2 q
q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -12 2 -8 -6 4 4 2 4 8 10 12
-1 + q - --- + q + q - -- + -- + q + 3 q + 7 q + 2 q + 5 q +
10 4 2
q q q
14 18 20 22
> 6 q + 3 q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 508]][a, z] |
Out[8]= | 2 2 2 4
-2 2 1 2 1 2 z 2 z z 2 2 4 2 z
-- + -- + ----- - ----- + ----- - 2 z - -- + ---- + -- + a z - 2 z + ---- -
4 2 6 2 4 2 2 2 6 4 2 4
a a a z a z a z a a a a
4 6
z 2 4 6 z
> -- + a z - z - --
2 2
a a |
In[9]:= | Kauffman[Link[11, Alternating, 508]][a, z] |
Out[9]= | 2 2
-2 3 2 1 2 1 2 2 2 z 2 z 5 z 14 z
-- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- - ---- - ----- -
6 4 2 6 2 4 2 2 2 5 3 5 3 6 4
a a a a z a z a z a z a z a a a a
2 3 3 3 4
11 z 2 2 2 z 2 z 10 z 3 3 3 4 17 z
> ----- + 2 a z + ---- + ---- + ----- + 4 a z - 2 a z + 4 z + ----- +
2 7 3 a 6
a a a a
4 4 5 5 5 5
47 z 42 z 2 4 4 4 3 z 10 z 14 z 16 z 5
> ----- + ----- - 7 a z + a z - ---- + ----- + ----- - ----- - 13 a z +
4 2 7 5 3 a
a a a a a
6 6 6 7 7 7
3 5 6 15 z 41 z 49 z 2 6 z 15 z 30 z
> 4 a z - 15 z - ----- - ----- - ----- + 8 a z + -- - ----- - ----- -
6 4 2 7 5 3
a a a a a a
7 8 8 8 9 9 9
3 z 7 8 4 z 7 z 14 z 5 z 12 z 7 z
> ---- + 11 a z + 11 z + ---- + ---- + ----- + ---- + ----- + ---- +
a 6 4 2 5 3 a
a a a a a
10 10
2 z 2 z
> ----- + -----
4 2
a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 10 1 3 1 5 3 8 5
-- + 9 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 10 q t +
q 9 4 7 3 5 3 5 2 3 2 3 q t
q t q t q t q t q t q t
3 3 2 5 2 5 3 7 3 7 4 9 4
> 9 q t + 10 q t + 12 q t + 8 q t + 8 q t + 5 q t + 9 q t +
9 5 11 5 11 6 13 6 15 7
> 3 q t + 4 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a508 |
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