 L11a529 |
 L11a531 |
| © | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: |
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 Knotscape |
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The 3-Component Link L11a530Visit L11a530's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X12,3,13,4 X20,14,21,13 X18,10,19,9 X10,22,11,21 X14,20,7,19 X16,5,17,6 X22,18,15,17 X2738 X4,11,5,12 X6,15,1,16 |
| Gauss Code: | {{1, -9, 2, -10, 7, -11}, {9, -1, 4, -5, 10, -2, 3, -6}, {11, -7, 8, -4, 6, -3, 5, -8}} |
| Jones Polynomial: | q-6 - 3q-5 + 9q-4 - 14q-3 + 20q-2 - 23q-1 + 24 - 20q + 16q2 - 9q3 + 4q4 - q5 |
| A2 (sl(3)) Invariant: | q-20 + q-18 - 2q-16 + 3q-14 + 4q-12 - 2q-10 + 7q-8 + 2q-6 + 2q-4 + 6q-2 - 1 + 6q2 - 3q4 + q6 + 4q8 - 4q10 + 2q12 + q14 - q16 |
| HOMFLY-PT Polynomial: | - a-4z2 + a-2 + a-2z2 + 2a-2z4 + z-2 + 1 - 2z2 - z4 - z6 - 2a2z-2 - 3a2 + a2z2 + 3a2z4 + a4z-2 - 3a4z2 + a6 |
| Kauffman Polynomial: | - a-5z3 + a-5z5 + 2a-4z2 - 5a-4z4 + 4a-4z6 + 5a-3z3 - 11a-3z5 + 8a-3z7 - a-2 - 2a-2z2 + 8a-2z4 - 14a-2z6 + 10a-2z8 + 6a-1z3 - 11a-1z5 - a-1z7 + 7a-1z9 - z-2 + 5 - 12z2 + 27z4 - 37z6 + 16z8 + 2z10 + 2az-1 - 9az + 18az3 - 11az5 - 13az7 + 12az9 - 2a2z-2 + 11a2 - 23a2z2 + 32a2z4 - 34a2z6 + 12a2z8 + 2a2z10 + 2a3z-1 - 9a3z + 21a3z3 - 18a3z5 - a3z7 + 5a3z9 - a4z-2 + 5a4 - 12a4z2 + 15a4z4 - 14a4z6 + 6a4z8 + 3a5z3 - 6a5z5 + 3a5z7 - a6 + 3a6z2 - 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, 530]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 530]] |
Out[4]= | PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 14, 21, 13], X[18, 10, 19, 9], > X[10, 22, 11, 21], X[14, 20, 7, 19], X[16, 5, 17, 6], X[22, 18, 15, 17], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 15, 1, 16]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 2, -10, 7, -11}, {9, -1, 4, -5, 10, -2, 3, -6},
> {11, -7, 8, -4, 6, -3, 5, -8}] |
In[6]:= | Jones[L][q] |
Out[6]= | -6 3 9 14 20 23 2 3 4 5
24 + q - -- + -- - -- + -- - -- - 20 q + 16 q - 9 q + 4 q - q
5 4 3 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -20 -18 2 3 4 2 7 2 2 6 2 4
-1 + q + q - --- + --- + --- - --- + -- + -- + -- + -- + 6 q - 3 q +
16 14 12 10 8 6 4 2
q q q q q q q q
6 8 10 12 14 16
> q + 4 q - 4 q + 2 q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 530]][a, z] |
Out[8]= | 2 4 2 2
-2 2 6 -2 2 a a 2 z z 2 2 4 2 4
1 + a - 3 a + a + z - ---- + -- - 2 z - -- + -- + a z - 3 a z - z +
2 2 4 2
z z a a
4
2 z 2 4 6
> ---- + 3 a z - z
2
a |
In[9]:= | Kauffman[Link[11, Alternating, 530]][a, z] |
Out[9]= | 2 4 3
-2 2 4 6 -2 2 a a 2 a 2 a 3
5 - a + 11 a + 5 a - a - z - ---- - -- + --- + ---- - 9 a z - 9 a z -
2 2 z z
z z
2 2 3 3 3
2 2 z 2 z 2 2 4 2 6 2 z 5 z 6 z
> 12 z + ---- - ---- - 23 a z - 12 a z + 3 a z - -- + ---- + ---- +
4 2 5 3 a
a a a a
4 4
3 3 3 5 3 4 5 z 8 z 2 4 4 4
> 18 a z + 21 a z + 3 a z + 27 z - ---- + ---- + 32 a z + 15 a z -
4 2
a a
5 5 5
6 4 z 11 z 11 z 5 3 5 5 5 6
> 3 a z + -- - ----- - ----- - 11 a z - 18 a z - 6 a z - 37 z +
5 3 a
a a
6 6 7 7
4 z 14 z 2 6 4 6 6 6 8 z z 7 3 7
> ---- - ----- - 34 a z - 14 a z + a z + ---- - -- - 13 a z - a z +
4 2 3 a
a a a
8 9
5 7 8 10 z 2 8 4 8 7 z 9 3 9
> 3 a z + 16 z + ----- + 12 a z + 6 a z + ---- + 12 a z + 5 a z +
2 a
a
10 2 10
> 2 z + 2 a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 13 1 1 3 6 3 8 6 12
-- + 13 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
8 11 12 3 3 2 5 2 5 3
> ----- + ---- + --- + 9 q t + 11 q t + 7 q t + 10 q t + 3 q t +
3 2 3 q t
q t q t
7 3 7 4 9 4 11 5
> 6 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a530 |
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