 L11a440 |
 L11a442 |
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The 3-Component Link L11a441Visit L11a441's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X14,3,15,4 X12,15,5,16 X22,17,13,18 X16,7,17,8 X8,22,9,21 X20,12,21,11 X18,10,19,9 X10,20,11,19 X2536 X4,13,1,14 |
| Gauss Code: | {{1, -10, 2, -11}, {10, -1, 5, -6, 8, -9, 7, -3}, {11, -2, 3, -5, 4, -8, 9, -7, 6, -4}} |
| Jones Polynomial: | - q-7 + 3q-6 - 6q-5 + 11q-4 - 14q-3 + 19q-2 - 18q-1 + 17 - 13q + 9q2 - 4q3 + q4 |
| A2 (sl(3)) Invariant: | - q-22 - q-20 + q-18 - q-16 + 4q-14 + 4q-12 + q-10 + 7q-8 + q-6 + 4q-4 + 3q-2 + 5q2 - 2q4 + 2q6 + q8 - 2q10 + q12 |
| HOMFLY-PT Polynomial: | a-2z2 + a-2z4 + z-2 + 4 + 3z2 - z4 - z6 - 2a2z-2 - 9a2 - 13a2z2 - 8a2z4 - 2a2z6 + a4z-2 + 7a4 + 9a4z2 + 3a4z4 - 2a6 - a6z2 |
| Kauffman Polynomial: | a-4z4 - a-3z3 + 4a-3z5 - a-2 + 4a-2z2 - 9a-2z4 + 9a-2z6 - a-1z + 7a-1z3 - 16a-1z5 + 12a-1z7 - z-2 + 3 - 2z4 - 10z6 + 10z8 + 2az-1 - 8az + 20az3 - 29az5 + 6az7 + 5az9 - 2a2z-2 + 11a2 - 29a2z2 + 47a2z4 - 48a2z6 + 16a2z8 + a2z10 + 2a3z-1 - 12a3z + 24a3z3 - 11a3z5 - 13a3z7 + 8a3z9 - a4z-2 + 11a4 - 34a4z2 + 55a4z4 - 41a4z6 + 9a4z8 + a4z10 - 7a5z + 17a5z3 - 6a5z5 - 6a5z7 + 3a5z9 + 3a6 - 9a6z2 + 16a6z4 - 12a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7 |
| Khovanov Homology: |
|
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, 441]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 441]] |
Out[4]= | PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[12, 15, 5, 16], X[22, 17, 13, 18], > X[16, 7, 17, 8], X[8, 22, 9, 21], X[20, 12, 21, 11], X[18, 10, 19, 9], > X[10, 20, 11, 19], X[2, 5, 3, 6], X[4, 13, 1, 14]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 8, -9, 7, -3},
> {11, -2, 3, -5, 4, -8, 9, -7, 6, -4}] |
In[6]:= | Jones[L][q] |
Out[6]= | -7 3 6 11 14 19 18 2 3 4
17 - q + -- - -- + -- - -- + -- - -- - 13 q + 9 q - 4 q + q
6 5 4 3 2 q
q q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -22 -20 -18 -16 4 4 -10 7 -6 4 3 2
-q - q + q - q + --- + --- + q + -- + q + -- + -- + 5 q -
14 12 8 4 2
q q q q q
4 6 8 10 12
> 2 q + 2 q + q - 2 q + q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 441]][a, z] |
Out[8]= | 2 4 2
2 4 6 -2 2 a a 2 z 2 2 4 2
4 - 9 a + 7 a - 2 a + z - ---- + -- + 3 z + -- - 13 a z + 9 a z -
2 2 2
z z a
4
6 2 4 z 2 4 4 4 6 2 6
> a z - z + -- - 8 a z + 3 a z - z - 2 a z
2
a |
In[9]:= | Kauffman[Link[11, Alternating, 441]][a, z] |
Out[9]= | 2 4 3
-2 2 4 6 -2 2 a a 2 a 2 a z
3 - a + 11 a + 11 a + 3 a - z - ---- - -- + --- + ---- - - - 8 a z -
2 2 z z a
z z
2 3
3 5 7 4 z 2 2 4 2 6 2 z
> 12 a z - 7 a z - 2 a z + ---- - 29 a z - 34 a z - 9 a z - -- +
2 3
a a
3 4 4
7 z 3 3 3 5 3 7 3 4 z 9 z
> ---- + 20 a z + 24 a z + 17 a z + 5 a z - 2 z + -- - ---- +
a 4 2
a a
5 5
2 4 4 4 6 4 4 z 16 z 5 3 5
> 47 a z + 55 a z + 16 a z + ---- - ----- - 29 a z - 11 a z -
3 a
a
6 7
5 5 7 5 6 9 z 2 6 4 6 6 6 12 z
> 6 a z - 4 a z - 10 z + ---- - 48 a z - 41 a z - 12 a z + ----- +
2 a
a
7 3 7 5 7 7 7 8 2 8 4 8
> 6 a z - 13 a z - 6 a z + a z + 10 z + 16 a z + 9 a z +
6 8 9 3 9 5 9 2 10 4 10
> 3 a z + 5 a z + 8 a z + 3 a z + a z + a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 10 1 2 1 4 2 7 5 8
-- + 10 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- +
q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3
q t q t q t q t q t q t q t q t
6 11 8 7 11 3 3 2 5 2
> ----- + ----- + ----- + ---- + --- + 6 q t + 7 q t + 3 q t + 6 q t +
5 3 5 2 3 2 3 q t
q t q t q t q t
5 3 7 3 9 4
> q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a441 |
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