 L11a472 |
 L11a474 |
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The 3-Component Link L11a473Visit L11a473's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X18,10,19,9 X8,18,9,17 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11 |
| Gauss Code: | {{1, -10, 2, -11}, {6, -5, 9, -7, 8, -4}, {10, -1, 3, -6, 5, -2, 11, -3, 7, -8, 4, -9}} |
| Jones Polynomial: | q-6 - 3q-5 + 7q-4 - 11q-3 + 16q-2 - 17q-1 + 19 - 15q + 12q2 - 7q3 + 3q4 - q5 |
| A2 (sl(3)) Invariant: | q-18 + q-14 + 3q-12 - q-10 + 5q-8 + 2q-6 + 3q-4 + 7q-2 + 1 + 7q2 - q4 + q6 + q8 - 3q10 + q12 - q14 |
| HOMFLY-PT Polynomial: | - 3a-2 - 6a-2z2 - 4a-2z4 - a-2z6 + z-2 + 10 + 18z2 + 15z4 + 6z6 + z8 - 2a2z-2 - 10a2 - 15a2z2 - 9a2z4 - 2a2z6 + a4z-2 + 3a4 + 3a4z2 + a4z4 |
| Kauffman Polynomial: | - 2a-5z3 + a-5z5 - 5a-4z4 + 3a-4z6 - 3a-3z + 10a-3z3 - 13a-3z5 + 6a-3z7 + 5a-2 - 18a-2z2 + 30a-2z4 - 22a-2z6 + 8a-2z8 - 10a-1z + 21a-1z3 - 7a-1z5 - 5a-1z7 + 5a-1z9 - z-2 + 15 - 48z2 + 72z4 - 49z6 + 14z8 + z10 + 2az-1 - 13az + 16az3 - 13az7 + 8az9 - 2a2z-2 + 13a2 - 33a2z2 + 40a2z4 - 32a2z6 + 10a2z8 + a2z10 + 2a3z-1 - 7a3z + 13a3z3 - 15a3z5 + a3z7 + 3a3z9 - a4z-2 + 3a4 - 7a4z6 + 4a4z8 - a5z + 6a5z3 - 8a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6 |
| 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, 473]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 473]] |
Out[4]= | PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[22, 15, 17, 16], > X[18, 10, 19, 9], X[8, 18, 9, 17], X[20, 13, 21, 14], X[14, 21, 15, 22], > X[16, 19, 5, 20], X[2, 5, 3, 6], X[4, 12, 1, 11]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {6, -5, 9, -7, 8, -4},
> {10, -1, 3, -6, 5, -2, 11, -3, 7, -8, 4, -9}] |
In[6]:= | Jones[L][q] |
Out[6]= | -6 3 7 11 16 17 2 3 4 5
19 + q - -- + -- - -- + -- - -- - 15 q + 12 q - 7 q + 3 q - q
5 4 3 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -18 -14 3 -10 5 2 3 7 2 4 6 8
1 + q + q + --- - q + -- + -- + -- + -- + 7 q - q + q + q -
12 8 6 4 2
q q q q q
10 12 14
> 3 q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 473]][a, z] |
Out[8]= | 2 4 2
3 2 4 -2 2 a a 2 6 z 2 2 4 2
10 - -- - 10 a + 3 a + z - ---- + -- + 18 z - ---- - 15 a z + 3 a z +
2 2 2 2
a z z a
4 6
4 4 z 2 4 4 4 6 z 2 6 8
> 15 z - ---- - 9 a z + a z + 6 z - -- - 2 a z + z
2 2
a a |
In[9]:= | Kauffman[Link[11, Alternating, 473]][a, z] |
Out[9]= | 2 4 3
5 2 4 6 -2 2 a a 2 a 2 a 3 z 10 z
15 + -- + 13 a + 3 a - a - z - ---- - -- + --- + ---- - --- - ---- -
2 2 2 z z 3 a
a z z a
2 3
3 5 2 18 z 2 2 6 2 2 z
> 13 a z - 7 a z - a z - 48 z - ----- - 33 a z + 3 a z - ---- +
2 5
a a
3 3 4 4
10 z 21 z 3 3 3 5 3 4 5 z 30 z
> ----- + ----- + 16 a z + 13 a z + 6 a z + 72 z - ---- + ----- +
3 a 4 2
a a a
5 5 5
2 4 6 4 z 13 z 7 z 3 5 5 5 6
> 40 a z - 3 a z + -- - ----- - ---- - 15 a z - 8 a z - 49 z +
5 3 a
a a
6 6 7 7
3 z 22 z 2 6 4 6 6 6 6 z 5 z 7 3 7
> ---- - ----- - 32 a z - 7 a z + a z + ---- - ---- - 13 a z + a z +
4 2 3 a
a a a
8 9
5 7 8 8 z 2 8 4 8 5 z 9 3 9
> 3 a z + 14 z + ---- + 10 a z + 4 a z + ---- + 8 a z + 3 a z +
2 a
a
10 2 10
> z + a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 10 1 2 1 5 2 6 5 10
-- + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- +
q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2
q t q t q t q t q t q t q t q t
8 9 8 3 3 2 5 2 5 3
> ----- + ---- + --- + 7 q t + 8 q t + 5 q t + 7 q t + 2 q t +
3 2 3 q t
q t q t
7 3 7 4 9 4 11 5
> 5 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a473 |
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