 L11a451 |
 L11a453 |
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The 3-Component Link L11a452Visit L11a452's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X14,4,15,3 X22,18,13,17 X16,8,17,7 X12,14,5,13 X8,21,9,22 X20,9,21,10 X18,11,19,12 X10,19,11,20 X2536 X4,16,1,15 |
| Gauss Code: | {{1, -10, 2, -11}, {10, -1, 4, -6, 7, -9, 8, -5}, {5, -2, 11, -4, 3, -8, 9, -7, 6, -3}} |
| Jones Polynomial: | q-5 - 2q-4 + 5q-3 - 6q-2 + 10q-1 - 11 + 12q - 10q2 + 9q3 - 6q4 + 3q5 - q6 |
| A2 (sl(3)) Invariant: | q-16 + q-14 + q-12 + 4q-10 + 3q-8 + 4q-6 + 5q-4 + q-2 + 3 + q2 + q4 + 3q6 - q8 + 2q10 - q12 - q14 + q16 - q18 |
| HOMFLY-PT Polynomial: | - a-4 - 2a-4z2 - a-4z4 + 2a-2 + 3a-2z2 + 3a-2z4 + a-2z6 + z-2 + 2 + 3z2 + 3z4 + z6 - 2a2z-2 - 5a2 - 6a2z2 - 2a2z4 + a4z-2 + 2a4 + a4z2 |
| Kauffman Polynomial: | a-7z3 + 3a-6z4 + a-5z - 4a-5z3 + 6a-5z5 - a-4 + 7a-4z2 - 14a-4z4 + 9a-4z6 + a-3z + 4a-3z3 - 16a-3z5 + 9a-3z7 + a-2z2 - 13a-2z6 + 7a-2z8 - 3a-1z + 12a-1z3 - 12a-1z5 - 5a-1z7 + 4a-1z9 - z-2 + 9 - 32z2 + 56z4 - 40z6 + 7z8 + z10 + 2az-1 - 8az + az3 + 24az5 - 24az7 + 6az9 - 2a2z-2 + 13a2 - 39a2z2 + 52a2z4 - 24a2z6 + a2z8 + a2z10 + 2a3z-1 - 5a3z - 2a3z3 + 14a3z5 - 10a3z7 + 2a3z9 - a4z-2 + 6a4 - 13a4z2 + 13a4z4 - 6a4z6 + a4z8 |
| 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, 452]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 452]] |
Out[4]= | PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[22, 18, 13, 17], X[16, 8, 17, 7], > X[12, 14, 5, 13], X[8, 21, 9, 22], X[20, 9, 21, 10], X[18, 11, 19, 12], > X[10, 19, 11, 20], X[2, 5, 3, 6], X[4, 16, 1, 15]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {10, -1, 4, -6, 7, -9, 8, -5},
> {5, -2, 11, -4, 3, -8, 9, -7, 6, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | -5 2 5 6 10 2 3 4 5 6
-11 + q - -- + -- - -- + -- + 12 q - 10 q + 9 q - 6 q + 3 q - q
4 3 2 q
q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -16 -14 -12 4 3 4 5 -2 2 4 6 8
3 + q + q + q + --- + -- + -- + -- + q + q + q + 3 q - q +
10 8 6 4
q q q q
10 12 14 16 18
> 2 q - q - q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 452]][a, z] |
Out[8]= | 2 4 2 2
-4 2 2 4 -2 2 a a 2 2 z 3 z 2 2
2 - a + -- - 5 a + 2 a + z - ---- + -- + 3 z - ---- + ---- - 6 a z +
2 2 2 4 2
a z z a a
4 4 6
4 2 4 z 3 z 2 4 6 z
> a z + 3 z - -- + ---- - 2 a z + z + --
4 2 2
a a a |
In[9]:= | Kauffman[Link[11, Alternating, 452]][a, z] |
Out[9]= | 2 4 3
-4 2 4 -2 2 a a 2 a 2 a z z 3 z
9 - a + 13 a + 6 a - z - ---- - -- + --- + ---- + -- + -- - --- - 8 a z -
2 2 z z 5 3 a
z z a a
2 2 3 3 3
3 2 7 z z 2 2 4 2 z 4 z 4 z
> 5 a z - 32 z + ---- + -- - 39 a z - 13 a z + -- - ---- + ---- +
4 2 7 5 3
a a a a a
3 4 4
12 z 3 3 3 4 3 z 14 z 2 4 4 4
> ----- + a z - 2 a z + 56 z + ---- - ----- + 52 a z + 13 a z +
a 6 4
a a
5 5 5 6 6
6 z 16 z 12 z 5 3 5 6 9 z 13 z
> ---- - ----- - ----- + 24 a z + 14 a z - 40 z + ---- - ----- -
5 3 a 4 2
a a a a
7 7 8
2 6 4 6 9 z 5 z 7 3 7 8 7 z
> 24 a z - 6 a z + ---- - ---- - 24 a z - 10 a z + 7 z + ---- +
3 a 2
a a
9
2 8 4 8 4 z 9 3 9 10 2 10
> a z + a z + ---- + 6 a z + 2 a z + z + a z
a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3 1 1 1 4 2 3 3 7
8 q + 7 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- +
11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2
q t q t q t q t q t q t q t q t
3 4 7 q 3 5 5 2 7 2 7 3
> ---- + --- + --- + 5 q t + 5 q t + 4 q t + 5 q t + 2 q t +
2 q t t
q t
9 3 9 4 11 4 13 5
> 4 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a452 |
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