 L11n367 |
 L11n369 |
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The 3-Component Link L11n368Visit L11n368's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X10,3,11,4 X7,14,8,15 X15,20,16,21 X11,19,12,18 X17,13,18,12 X19,22,20,17 X21,16,22,5 X13,8,14,9 X2536 X4,9,1,10 |
| Gauss Code: | {{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8}} |
| Jones Polynomial: | - q-10 + 2q-9 - 2q-8 + 2q-7 + q-4 - q-3 + 3q-2 - q-1 + 1 |
| A2 (sl(3)) Invariant: | - q-32 - q-30 + q-28 + 2q-24 + q-22 + q-20 + 2q-18 + q-16 + 4q-14 + 2q-12 + 4q-10 + 4q-8 + 3q-6 + 2q-4 + q-2 + 1 |
| HOMFLY-PT Polynomial: | a2z-2 + 4a2 + 4a2z2 + a2z4 - 2a4z-2 - 6a4 - 7a4z2 - 5a4z4 - a4z6 + a6z-2 + a6 + a6z2 + 2a8 + 2a8z2 - a10 |
| Kauffman Polynomial: | a2z-2 - 5a2 + 8a2z2 - 5a2z4 + a2z6 - 2a3z-1 + 6a3z - 2a3z3 - 3a3z5 + a3z7 + 2a4z-2 - 8a4 + 15a4z2 - 14a4z4 + 3a4z6 - 2a5z-1 + 8a5z - 8a5z3 + a5z5 + a6z-2 - a6 - 6a6z2 + 11a6z4 - 7a6z6 + a6z8 - a7z3 + 7a7z5 - 6a7z7 + a7z9 + 5a8 - 22a8z2 + 36a8z4 - 20a8z6 + 3a8z8 - 4a9z + 11a9z3 - 2a9z5 - 4a9z7 + a9z9 + 2a10 - 9a10z2 + 16a10z4 - 11a10z6 + 2a10z8 - 2a11z + 6a11z3 - 5a11z5 + a11z7 |
| 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, 368]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 368]] |
Out[4]= | PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[7, 14, 8, 15], X[15, 20, 16, 21], > X[11, 19, 12, 18], X[17, 13, 18, 12], X[19, 22, 20, 17], X[21, 16, 22, 5], > X[13, 8, 14, 9], X[2, 5, 3, 6], X[4, 9, 1, 10]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7},
> {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8}] |
In[6]:= | Jones[L][q] |
Out[6]= | -10 2 2 2 -4 -3 3 1
1 - q + -- - -- + -- + q - q + -- - -
9 8 7 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -32 -30 -28 2 -22 -20 2 -16 4 2 4
1 - q - q + q + --- + q + q + --- + q + --- + --- + --- +
24 18 14 12 10
q q q q q
4 3 2 -2
> -- + -- + -- + q
8 6 4
q q q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 368]][a, z] |
Out[8]= | 2 4 6
2 4 6 8 10 a 2 a a 2 2 4 2 6 2
4 a - 6 a + a + 2 a - a + -- - ---- + -- + 4 a z - 7 a z + a z +
2 2 2
z z z
8 2 2 4 4 4 4 6
> 2 a z + a z - 5 a z - a z |
In[9]:= | Kauffman[Link[11, NonAlternating, 368]][a, z] |
Out[9]= | 2 4 6 3 5
2 4 6 8 10 a 2 a a 2 a 2 a 3
-5 a - 8 a - a + 5 a + 2 a + -- + ---- + -- - ---- - ---- + 6 a z +
2 2 2 z z
z z z
5 9 11 2 2 4 2 6 2 8 2
> 8 a z - 4 a z - 2 a z + 8 a z + 15 a z - 6 a z - 22 a z -
10 2 3 3 5 3 7 3 9 3 11 3 2 4
> 9 a z - 2 a z - 8 a z - a z + 11 a z + 6 a z - 5 a z -
4 4 6 4 8 4 10 4 3 5 5 5 7 5
> 14 a z + 11 a z + 36 a z + 16 a z - 3 a z + a z + 7 a z -
9 5 11 5 2 6 4 6 6 6 8 6 10 6
> 2 a z - 5 a z + a z + 3 a z - 7 a z - 20 a z - 11 a z +
3 7 7 7 9 7 11 7 6 8 8 8 10 8 7 9
> a z - 6 a z - 4 a z + a z + a z + 3 a z + 2 a z + a z +
9 9
> a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 3 1 1 1 1 1 2 1
-- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ +
5 3 21 9 19 8 17 8 17 7 15 7 15 6 13 6
q q q t q t q t q t q t q t q t
1 1 2 2 4 2 2 2
> ------ + ------ + ------ + ------ + ------ + ----- + ------ + ----- +
15 5 13 5 11 5 13 4 11 4 9 4 11 3 9 3
q t q t q t q t q t q t q t q t
1 1 2 1 1 1 t 2
> ----- + ----- + ----- + ----- + ---- + ---- + -- + q t
7 3 9 2 7 2 5 2 7 5 3
q t q t q t q t q t q t q |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n368 |
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