 L11a512 |
 L11a514 |
| © | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: |
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The 3-Component Link L11a513Visit L11a513's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X22,18,13,17 X10,20,11,19 X18,10,19,9 X20,12,21,11 X12,22,7,21 X2738 X6,13,1,14 |
| Gauss Code: | {{1, -10, 3, -4, 2, -11}, {10, -1, 7, -6, 8, -9}, {11, -3, 4, -2, 5, -7, 6, -8, 9, -5}} |
| Jones Polynomial: | q-6 - 2q-5 + 5q-4 - 7q-3 + 11q-2 - 12q-1 + 13 - 11q + 9q2 - 5q3 + 3q4 - q5 |
| A2 (sl(3)) Invariant: | q-18 + q-16 + q-14 + 3q-12 + q-10 + 5q-8 + 3q-6 + 2q-4 + 4q-2 - 1 + 4q2 + 2q6 + 2q8 - q10 + q12 - q14 |
| HOMFLY-PT Polynomial: | - 4a-2z2 - 4a-2z4 - a-2z6 + z-2 + 5 + 12z2 + 13z4 + 6z6 + z8 - 2a2z-2 - 9a2 - 16a2z2 - 10a2z4 - 2a2z6 + a4z-2 + 4a4 + 4a4z2 + a4z4 |
| Kauffman Polynomial: | - 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 + 3a-3z3 - 8a-3z5 + 4a-3z7 - a-2 - a-2z2 + 6a-2z4 - 8a-2z6 + 4a-2z8 - 2a-1z3 + 7a-1z5 - 6a-1z7 + 3a-1z9 - z-2 + 5 - 14z2 + 24z4 - 14z6 + 3z8 + z10 + 2az-1 - 9az + 4az3 + 14az5 - 14az7 + 5az9 - 2a2z-2 + 11a2 - 25a2z2 + 27a2z4 - 14a2z6 + 2a2z8 + a2z10 + 2a3z-1 - 9a3z + 14a3z3 - 8a3z5 - 2a3z7 + 2a3z9 - a4z-2 + 5a4 - 11a4z2 + 12a4z4 - 10a4z6 + 3a4z8 + 3a5z3 - 6a5z5 + 2a5z7 - a6 + 4a6z2 - 4a6z4 + 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, 513]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 513]] |
Out[4]= | PD[X[8, 1, 9, 2], X[16, 5, 17, 6], X[14, 3, 15, 4], X[4, 15, 5, 16], > X[22, 18, 13, 17], X[10, 20, 11, 19], X[18, 10, 19, 9], X[20, 12, 21, 11], > X[12, 22, 7, 21], X[2, 7, 3, 8], X[6, 13, 1, 14]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 3, -4, 2, -11}, {10, -1, 7, -6, 8, -9},
> {11, -3, 4, -2, 5, -7, 6, -8, 9, -5}] |
In[6]:= | Jones[L][q] |
Out[6]= | -6 2 5 7 11 12 2 3 4 5
13 + q - -- + -- - -- + -- - -- - 11 q + 9 q - 5 q + 3 q - q
5 4 3 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -18 -16 -14 3 -10 5 3 2 4 2 6 8
-1 + q + q + q + --- + q + -- + -- + -- + -- + 4 q + 2 q + 2 q -
12 8 6 4 2
q q q q q
10 12 14
> q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 513]][a, z] |
Out[8]= | 2 4 2
2 4 -2 2 a a 2 4 z 2 2 4 2 4
5 - 9 a + 4 a + z - ---- + -- + 12 z - ---- - 16 a z + 4 a z + 13 z -
2 2 2
z z a
4 6
4 z 2 4 4 4 6 z 2 6 8
> ---- - 10 a z + a z + 6 z - -- - 2 a z + z
2 2
a a |
In[9]:= | Kauffman[Link[11, Alternating, 513]][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 3 z z 2 2 4 2 6 2 2 z 3 z 2 z
> 14 z + ---- - -- - 25 a z - 11 a z + 4 a z - ---- + ---- - ---- +
4 2 5 3 a
a a a a
4 4
3 3 3 5 3 4 7 z 6 z 2 4 4 4
> 4 a z + 14 a z + 3 a z + 24 z - ---- + ---- + 27 a z + 12 a z -
4 2
a a
5 5 5 6
6 4 z 8 z 7 z 5 3 5 5 5 6 3 z
> 4 a z + -- - ---- + ---- + 14 a z - 8 a z - 6 a z - 14 z + ---- -
5 3 a 4
a a a
6 7 7
8 z 2 6 4 6 6 6 4 z 6 z 7 3 7
> ---- - 14 a z - 10 a z + a z + ---- - ---- - 14 a z - 2 a z +
2 3 a
a a
8 9
5 7 8 4 z 2 8 4 8 3 z 9 3 9 10
> 2 a z + 3 z + ---- + 2 a z + 3 a z + ---- + 5 a z + 2 a z + z +
2 a
a
2 10
> a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 8 1 1 2 3 2 4 3 7
- + 7 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
4 5 7 3 3 2 5 2 5 3
> ----- + ---- + --- + 5 q t + 6 q t + 4 q t + 6 q t + 2 q t +
3 2 3 q t
q t q t
7 3 7 4 9 4 11 5
> 3 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a513 |
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