 L11a527 |
 L11a529 |
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The 3-Component Link L11a528Visit L11a528's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X14,3,15,4 X22,18,13,17 X16,11,17,12 X12,15,7,16 X10,22,11,21 X20,10,21,9 X18,6,19,5 X2738 X4,13,5,14 X6,20,1,19 |
| Gauss Code: | {{1, -9, 2, -10, 8, -11}, {9, -1, 7, -6, 4, -5}, {10, -2, 5, -4, 3, -8, 11, -7, 6, -3}} |
| Jones Polynomial: | q-5 - 3q-4 + 7q-3 - 9q-2 + 14q-1 - 14 + 15q - 13q2 + 10q3 - 6q4 + 3q5 - q6 |
| A2 (sl(3)) Invariant: | q-16 + q-14 + 4q-10 + 2q-8 + 4q-6 + 6q-4 + 2q-2 + 6 + q2 + q4 + q6 - 3q8 + 2q10 - q12 + q16 - q18 |
| HOMFLY-PT Polynomial: | - 2a-4z2 - a-4z4 - 2a-2 - a-2z2 + 2a-2z4 + a-2z6 + z-2 + 7 + 11z2 + 8z4 + 2z6 - 2a2z-2 - 7a2 - 9a2z2 - 3a2z4 + a4z-2 + 2a4 + a4z2 |
| Kauffman Polynomial: | a-7z3 + 3a-6z4 - 3a-5z3 + 6a-5z5 + 4a-4z2 - 14a-4z4 + 10a-4z6 - 2a-3z + 14a-3z3 - 28a-3z5 + 13a-3z7 + 4a-2 - 14a-2z2 + 28a-2z4 - 36a-2z6 + 13a-2z8 - 6a-1z + 20a-1z3 - 8a-1z5 - 17a-1z7 + 8a-1z9 - z-2 + 13 - 54z2 + 98z4 - 69z6 + 11z8 + 2z10 + 2az-1 - 7az - 5az3 + 46az5 - 44az7 + 11az9 - 2a2z-2 + 13a2 - 46a2z2 + 63a2z4 - 28a2z6 - a2z8 + 2a2z10 + 2a3z-1 - 3a3z - 7a3z3 + 20a3z5 - 14a3z7 + 3a3z9 - a4z-2 + 5a4 - 10a4z2 + 10a4z4 - 5a4z6 + a4z8 |
| 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, 528]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 528]] |
Out[4]= | PD[X[8, 1, 9, 2], X[14, 3, 15, 4], X[22, 18, 13, 17], X[16, 11, 17, 12], > X[12, 15, 7, 16], X[10, 22, 11, 21], X[20, 10, 21, 9], X[18, 6, 19, 5], > X[2, 7, 3, 8], X[4, 13, 5, 14], X[6, 20, 1, 19]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 2, -10, 8, -11}, {9, -1, 7, -6, 4, -5},
> {10, -2, 5, -4, 3, -8, 11, -7, 6, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | -5 3 7 9 14 2 3 4 5 6
-14 + q - -- + -- - -- + -- + 15 q - 13 q + 10 q - 6 q + 3 q - q
4 3 2 q
q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -16 -14 4 2 4 6 2 2 4 6 8 10 12
6 + q + q + --- + -- + -- + -- + -- + q + q + q - 3 q + 2 q - q +
10 8 6 4 2
q q q q q
16 18
> q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 528]][a, z] |
Out[8]= | 2 4 2 2
2 2 4 -2 2 a a 2 2 z z 2 2 4 2
7 - -- - 7 a + 2 a + z - ---- + -- + 11 z - ---- - -- - 9 a z + a z +
2 2 2 4 2
a z z a a
4 4 6
4 z 2 z 2 4 6 z
> 8 z - -- + ---- - 3 a z + 2 z + --
4 2 2
a a a |
In[9]:= | Kauffman[Link[11, Alternating, 528]][a, z] |
Out[9]= | 2 4 3
4 2 4 -2 2 a a 2 a 2 a 2 z 6 z
13 + -- + 13 a + 5 a - z - ---- - -- + --- + ---- - --- - --- - 7 a z -
2 2 2 z z 3 a
a z z a
2 2 3 3 3
3 2 4 z 14 z 2 2 4 2 z 3 z 14 z
> 3 a z - 54 z + ---- - ----- - 46 a z - 10 a z + -- - ---- + ----- +
4 2 7 5 3
a a a a a
3 4 4 4
20 z 3 3 3 4 3 z 14 z 28 z 2 4
> ----- - 5 a z - 7 a z + 98 z + ---- - ----- + ----- + 63 a z +
a 6 4 2
a a a
5 5 5 6
4 4 6 z 28 z 8 z 5 3 5 6 10 z
> 10 a z + ---- - ----- - ---- + 46 a z + 20 a z - 69 z + ----- -
5 3 a 4
a a a
6 7 7
36 z 2 6 4 6 13 z 17 z 7 3 7 8
> ----- - 28 a z - 5 a z + ----- - ----- - 44 a z - 14 a z + 11 z +
2 3 a
a a
8 9
13 z 2 8 4 8 8 z 9 3 9 10 2 10
> ----- - a z + a z + ---- + 11 a z + 3 a z + 2 z + 2 a z
2 a
a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3 1 2 1 5 2 4 5 10
9 q + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- +
11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2
q t q t q t q t q t q t q t q t
6 6 8 q 3 5 5 2 7 2 7 3
> ---- + --- + --- + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t +
2 q t t
q t
9 3 9 4 11 4 13 5
> 4 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a528 |
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