 L11n302 |
 L11n304 |
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The 3-Component Link L11n303Visit L11n303's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,3,13,4 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X21,15,22,14 X2536 X4,11,1,12 |
| Gauss Code: | {{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}} |
| Jones Polynomial: | q-2 + 1 + 2q - 2q2 + 4q3 - 4q4 + 4q5 - 3q6 + 2q7 - q8 |
| A2 (sl(3)) Invariant: | q-6 + 2q-4 + 4q-2 + 4 + 6q2 + 4q4 + 4q6 + 4q8 + q10 + 2q12 - q14 - q18 - 2q20 - q24 |
| HOMFLY-PT Polynomial: | - a-6z-2 - 3a-6 - 3a-6z2 - a-6z4 + 4a-4z-2 + 13a-4 + 14a-4z2 + 6a-4z4 + a-4z6 - 5a-2z-2 - 17a-2 - 19a-2z2 - 8a-2z4 - a-2z6 + 2z-2 + 7 + 5z2 + z4 |
| Kauffman Polynomial: | a-9z - 3a-9z3 + a-9z5 - a-8 + 3a-8z2 - 6a-8z4 + 2a-8z6 + a-7z-1 - 2a-7z + 4a-7z3 - 6a-7z5 + 2a-7z7 - a-6z-2 + 3a-6z2 - a-6z4 - 2a-6z6 + a-6z8 + 5a-5z-1 - 19a-5z + 26a-5z3 - 14a-5z5 + 3a-5z7 - 4a-4z-2 + 13a-4 - 16a-4z2 + 15a-4z4 - 6a-4z6 + a-4z8 + 9a-3z-1 - 35a-3z + 41a-3z3 - 16a-3z5 + 2a-3z7 - 5a-2z-2 + 22a-2 - 39a-2z2 + 31a-2z4 - 10a-2z6 + a-2z8 + 5a-1z-1 - 19a-1z + 22a-1z3 - 9a-1z5 + a-1z7 - 2z-2 + 11 - 23z2 + 21z4 - 8z6 + z8 |
| 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, 303]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 303]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 21, 14, 20], X[19, 11, 20, 22], > X[15, 5, 16, 10], X[17, 9, 18, 8], X[7, 17, 8, 16], X[9, 19, 10, 18], > X[21, 15, 22, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {10, -1, -7, 6, -8, 5},
> {11, -2, -3, 9, -5, 7, -6, 8, -4, 3, -9, 4}] |
In[6]:= | Jones[L][q] |
Out[6]= | -2 2 3 4 5 6 7 8 1 + q + 2 q - 2 q + 4 q - 4 q + 4 q - 3 q + 2 q - q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -6 2 4 2 4 6 8 10 12 14 18
4 + q + -- + -- + 6 q + 4 q + 4 q + 4 q + q + 2 q - q - q -
4 2
q q
20 24
> 2 q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 303]][a, z] |
Out[8]= | 2 2 2
3 13 17 2 1 4 5 2 3 z 14 z 19 z
7 - -- + -- - -- + -- - ----- + ----- - ----- + 5 z - ---- + ----- - ----- +
6 4 2 2 6 2 4 2 2 2 6 4 2
a a a z a z a z a z a a a
4 4 4 6 6
4 z 6 z 8 z z z
> z - -- + ---- - ---- + -- - --
6 4 2 4 2
a a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 303]][a, z] |
Out[9]= | -8 13 22 2 1 4 5 1 5 9 5
11 - a + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- +
4 2 2 6 2 4 2 2 2 7 5 3 a z
a a z a z a z a z a z a z a z
2 2 2 2
z 2 z 19 z 35 z 19 z 2 3 z 3 z 16 z 39 z
> -- - --- - ---- - ---- - ---- - 23 z + ---- + ---- - ----- - ----- -
9 7 5 3 a 8 6 4 2
a a a a a a a a
3 3 3 3 3 4 4 4 4
3 z 4 z 26 z 41 z 22 z 4 6 z z 15 z 31 z
> ---- + ---- + ----- + ----- + ----- + 21 z - ---- - -- + ----- + ----- +
9 7 5 3 a 8 6 4 2
a a a a a a a a
5 5 5 5 5 6 6 6 6
z 6 z 14 z 16 z 9 z 6 2 z 2 z 6 z 10 z
> -- - ---- - ----- - ----- - ---- - 8 z + ---- - ---- - ---- - ----- +
9 7 5 3 a 8 6 4 2
a a a a a a a a
7 7 7 7 8 8 8
2 z 3 z 2 z z 8 z z z
> ---- + ---- + ---- + -- + z + -- + -- + --
7 5 3 a 6 4 2
a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3
3 1 1 1 q 3 5 7 5 2
3 q + 2 q + ----- + ----- + ---- + -- + q t + 2 q t + q t + 4 q t +
5 4 3 4 2 t
q t q t q t
7 2 7 3 9 3 9 4 11 4 11 5 13 5
> 3 q t + 2 q t + 2 q t + 2 q t + 2 q t + q t + 2 q t +
13 6 15 6 17 7
> q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n303 |
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