 L11a398 |
 L11a400 |
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The 3-Component Link L11a399Visit L11a399's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X18,22,19,21 X20,9,21,10 X10,19,5,20 X16,12,17,11 X22,18,11,17 |
| Gauss Code: | {{1, -4, 3, -6}, {2, -1, 5, -3, 8, -9}, {10, -2, 4, -5, 6, -10, 11, -7, 9, -8, 7, -11}} |
| Jones Polynomial: | - q-5 + 4q-4 - 7q-3 + 14q-2 - 17q-1 + 21 - 20q + 18q2 - 13q3 + 8q4 - 4q5 + q6 |
| A2 (sl(3)) Invariant: | - q-16 + q-14 + 3q-12 + 7q-8 + 6q-6 + 2q-4 + 7q-2 - 1 + 3q2 - q6 + 4q8 - 4q10 + q12 + q14 - 2q16 + q18 |
| HOMFLY-PT Polynomial: | a-4z2 + a-4z4 - 2a-2z2 - 2a-2z4 - a-2z6 + z-2 + 1 - z4 - z6 - 2a2z-2 - a2 + 2a2z2 + 2a2z4 + a4z-2 - a4z2 |
| Kauffman Polynomial: | - 2a-6z4 + a-6z6 + 5a-5z3 - 10a-5z5 + 4a-5z7 - 4a-4z2 + 15a-4z4 - 19a-4z6 + 7a-4z8 + 3a-3z3 + 2a-3z5 - 11a-3z7 + 6a-3z9 - 8a-2z2 + 30a-2z4 - 32a-2z6 + 9a-2z8 + 2a-2z10 - 6a-1z3 + 17a-1z5 - 22a-1z7 + 11a-1z9 - z-2 + 1 + 8z4 - 17z6 + 8z8 + 2z10 + 2az-1 - az - 2az3 - 3az5 - az7 + 5az9 - 2a2z-2 + a2 + 8a2z2 - 12a2z4 - a2z6 + 6a2z8 + 2a3z-1 - a3z + a3z3 - 7a3z5 + 6a3z7 - a4z-2 + a4 + 4a4z2 - 7a4z4 + 4a4z6 - a5z3 + a5z5 |
| 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, 399]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 399]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 6, 13, 5], X[8, 4, 9, 3], X[2, 14, 3, 13], > X[14, 7, 15, 8], X[4, 15, 1, 16], X[18, 22, 19, 21], X[20, 9, 21, 10], > X[10, 19, 5, 20], X[16, 12, 17, 11], X[22, 18, 11, 17]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -4, 3, -6}, {2, -1, 5, -3, 8, -9},
> {10, -2, 4, -5, 6, -10, 11, -7, 9, -8, 7, -11}] |
In[6]:= | Jones[L][q] |
Out[6]= | -5 4 7 14 17 2 3 4 5 6
21 - q + -- - -- + -- - -- - 20 q + 18 q - 13 q + 8 q - 4 q + q
4 3 2 q
q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -16 -14 3 7 6 2 7 2 6 8 10 12
-1 - q + q + --- + -- + -- + -- + -- + 3 q - q + 4 q - 4 q + q +
12 8 6 4 2
q q q q q
14 16 18
> q - 2 q + q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 399]][a, z] |
Out[8]= | 2 4 2 2 4 4
2 -2 2 a a z 2 z 2 2 4 2 4 z 2 z
1 - a + z - ---- + -- + -- - ---- + 2 a z - a z - z + -- - ---- +
2 2 4 2 4 2
z z a a a a
6
2 4 6 z
> 2 a z - z - --
2
a |
In[9]:= | Kauffman[Link[11, Alternating, 399]][a, z] |
Out[9]= | 2 4 3 2 2
2 4 -2 2 a a 2 a 2 a 3 4 z 8 z
1 + a + a - z - ---- - -- + --- + ---- - a z - a z - ---- - ---- +
2 2 z z 4 2
z z a a
3 3 3
2 2 4 2 5 z 3 z 6 z 3 3 3 5 3 4
> 8 a z + 4 a z + ---- + ---- - ---- - 2 a z + a z - a z + 8 z -
5 3 a
a a
4 4 4 5 5 5
2 z 15 z 30 z 2 4 4 4 10 z 2 z 17 z 5
> ---- + ----- + ----- - 12 a z - 7 a z - ----- + ---- + ----- - 3 a z -
6 4 2 5 3 a
a a a a a
6 6 6 7
3 5 5 5 6 z 19 z 32 z 2 6 4 6 4 z
> 7 a z + a z - 17 z + -- - ----- - ----- - a z + 4 a z + ---- -
6 4 2 5
a a a a
7 7 8 8 9
11 z 22 z 7 3 7 8 7 z 9 z 2 8 6 z
> ----- - ----- - a z + 6 a z + 8 z + ---- + ---- + 6 a z + ---- +
3 a 4 2 3
a a a a
9 10
11 z 9 10 2 z
> ----- + 5 a z + 2 z + -----
a 2
a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 12 1 3 1 4 3 10 6 9
-- + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- +
q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3
q t q t q t q t q t q t q t q t
8 3 3 2 5 2 5 3 7 3 7 4
> --- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + 3 q t +
q t
9 4 9 5 11 5 13 6
> 5 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a399 |
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