 L11a522 |
 L11a524 |
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 Knotscape |
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The 3-Component Link L11a523Visit L11a523's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X14,4,15,3 X18,7,19,8 X22,9,13,10 X10,21,11,22 X20,14,21,13 X12,17,7,18 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,20,1,19 |
| Gauss Code: | {{1, -9, 2, -10, 8, -11}, {3, -1, 4, -5, 9, -7}, {6, -2, 10, -8, 7, -3, 11, -6, 5, -4}} |
| Jones Polynomial: | q-6 - 3q-5 + 8q-4 - 12q-3 + 18q-2 - 20q-1 + 21 - 18q + 14q2 - 8q3 + 4q4 - q5 |
| A2 (sl(3)) Invariant: | q-18 + q-14 + 4q-12 + 7q-8 + 3q-6 + 2q-4 + 5q-2 - 3 + 5q2 - 2q4 + 2q6 + 3q8 - 2q10 + 2q12 - q14 |
| HOMFLY-PT Polynomial: | a-2 - 2a-2z2 - 3a-2z4 - a-2z6 + z-2 + 2 + 6z2 + 9z4 + 5z6 + z8 - 2a2z-2 - 6a2 - 11a2z2 - 8a2z4 - 2a2z6 + a4z-2 + 3a4 + 3a4z2 + a4z4 |
| Kauffman Polynomial: | - a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 + 3a-3z3 - 10a-3z5 + 7a-3z7 - 2a-2 + 3a-2z2 + a-2z4 - 10a-2z6 + 8a-2z8 + 2a-1z3 - 3a-1z5 - 4a-1z7 + 6a-1z9 - z-2 + 3 - 5z2 + 20z4 - 26z6 + 10z8 + 2z10 + 2az-1 - 9az + 8az3 + 10az5 - 21az7 + 11az9 - 2a2z-2 + 11a2 - 27a2z2 + 39a2z4 - 31a2z6 + 8a2z8 + 2a2z10 + 2a3z-1 - 9a3z + 13a3z3 - 5a3z5 - 7a3z7 + 5a3z9 - a4z-2 + 7a4 - 19a4z2 + 23a4z4 - 18a4z6 + 6a4z8 + 3a5z3 - 7a5z5 + 3a5z7 + 2a6z2 - 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, 523]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 523]] |
Out[4]= | PD[X[8, 1, 9, 2], X[14, 4, 15, 3], X[18, 7, 19, 8], X[22, 9, 13, 10], > X[10, 21, 11, 22], X[20, 14, 21, 13], X[12, 17, 7, 18], X[16, 6, 17, 5], > X[2, 11, 3, 12], X[4, 16, 5, 15], X[6, 20, 1, 19]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 2, -10, 8, -11}, {3, -1, 4, -5, 9, -7},
> {6, -2, 10, -8, 7, -3, 11, -6, 5, -4}] |
In[6]:= | Jones[L][q] |
Out[6]= | -6 3 8 12 18 20 2 3 4 5
21 + q - -- + -- - -- + -- - -- - 18 q + 14 q - 8 q + 4 q - q
5 4 3 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -18 -14 4 7 3 2 5 2 4 6 8
-3 + q + q + --- + -- + -- + -- + -- + 5 q - 2 q + 2 q + 3 q -
12 8 6 4 2
q q q q q
10 12 14
> 2 q + 2 q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 523]][a, z] |
Out[8]= | 2 4 2
-2 2 4 -2 2 a a 2 2 z 2 2 4 2
2 + a - 6 a + 3 a + z - ---- + -- + 6 z - ---- - 11 a z + 3 a z +
2 2 2
z z a
4 6
4 3 z 2 4 4 4 6 z 2 6 8
> 9 z - ---- - 8 a z + a z + 5 z - -- - 2 a z + z
2 2
a a |
In[9]:= | Kauffman[Link[11, Alternating, 523]][a, z] |
Out[9]= | 2 4 3
2 2 4 -2 2 a a 2 a 2 a 3 2
3 - -- + 11 a + 7 a - z - ---- - -- + --- + ---- - 9 a z - 9 a z - 5 z +
2 2 2 z z
a z z
2 2 3 3 3
2 z 3 z 2 2 4 2 6 2 z 3 z 2 z 3
> ---- + ---- - 27 a z - 19 a z + 2 a z - -- + ---- + ---- + 8 a z +
4 2 5 3 a
a a a a
4 4
3 3 5 3 4 6 z z 2 4 4 4 6 4
> 13 a z + 3 a z + 20 z - ---- + -- + 39 a z + 23 a z - 3 a z +
4 2
a a
5 5 5 6 6
z 10 z 3 z 5 3 5 5 5 6 4 z 10 z
> -- - ----- - ---- + 10 a z - 5 a z - 7 a z - 26 z + ---- - ----- -
5 3 a 4 2
a a a a
7 7
2 6 4 6 6 6 7 z 4 z 7 3 7 5 7
> 31 a z - 18 a z + a z + ---- - ---- - 21 a z - 7 a z + 3 a z +
3 a
a
8 9
8 8 z 2 8 4 8 6 z 9 3 9 10
> 10 z + ---- + 8 a z + 6 a z + ---- + 11 a z + 5 a z + 2 z +
2 a
a
2 10
> 2 a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 12 1 1 3 5 3 7 5 11
-- + 11 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- +
q 13 6 11 6 11 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
7 9 11 3 3 2 5 2 5 3
> ----- + ---- + --- + 8 q t + 10 q t + 6 q t + 9 q t + 3 q t +
3 2 3 q t
q t q t
7 3 7 4 9 4 11 5
> 5 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a523 |
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