 L11a474 |
 L11a476 |
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The 3-Component Link L11a475Visit L11a475's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X10,3,11,4 X14,8,15,7 X20,16,21,15 X18,11,19,12 X12,17,13,18 X22,20,17,19 X16,22,5,21 X8,14,9,13 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-6 - 2q-5 + 7q-4 - 11q-3 + 18q-2 - 20q-1 + 21 - 19q + 15q2 - 9q3 + 4q4 - q5 |
| A2 (sl(3)) Invariant: | q-20 + 2q-18 + q-16 + 5q-14 + 6q-12 + 7q-8 + 2q-6 + 4q-2 - 3 + 4q2 - 3q4 + 3q8 - 4q10 + 2q12 + q14 - q16 |
| HOMFLY-PT Polynomial: | - a-4z2 + a-2z2 + 2a-2z4 - z2 - z4 - z6 + a2z-2 + 3a2 + 4a2z2 + 3a2z4 - 2a4z-2 - 4a4 - 3a4z2 + a6z-2 + a6 |
| Kauffman Polynomial: | - a-5z3 + a-5z5 + 2a-4z2 - 5a-4z4 + 4a-4z6 - 2a-3z + 7a-3z3 - 12a-3z5 + 8a-3z7 + a-2 - 2a-2z2 + 3a-2z4 - 11a-2z6 + 9a-2z8 - 6a-1z + 22a-1z3 - 28a-1z5 + 7a-1z7 + 5a-1z9 + 3 - 14z2 + 31z4 - 38z6 + 17z8 + z10 - 6az + 18az3 - 18az5 - 3az7 + 8az9 + a2z-2 - a2 - 7a2z2 + 22a2z4 - 27a2z6 + 11a2z8 + a2z10 - 2a3z-1 + 2a3z + 4a3z3 - 7a3z5 + 3a3z9 + 2a4z-2 - 6a4 + 9a4z2 - 5a4z4 - 3a4z6 + 3a4z8 - 2a5z-1 + 4a5z - 4a5z5 + 2a5z7 + a6z-2 - 4a6 + 6a6z2 - 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, 475]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 475]] |
Out[4]= | PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[20, 16, 21, 15], > X[18, 11, 19, 12], X[12, 17, 13, 18], X[22, 20, 17, 19], X[16, 22, 5, 21], > X[8, 14, 9, 13], 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]= | -6 2 7 11 18 20 2 3 4 5
21 + q - -- + -- - -- + -- - -- - 19 q + 15 q - 9 q + 4 q - q
5 4 3 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -20 2 -16 5 6 7 2 4 2 4 8
-3 + q + --- + q + --- + --- + -- + -- + -- + 4 q - 3 q + 3 q -
18 14 12 8 6 2
q q q q q q
10 12 14 16
> 4 q + 2 q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 475]][a, z] |
Out[8]= | 2 4 6 2 2
2 4 6 a 2 a a 2 z z 2 2 4 2 4
3 a - 4 a + a + -- - ---- + -- - z - -- + -- + 4 a z - 3 a z - z +
2 2 2 4 2
z z z a a
4
2 z 2 4 6
> ---- + 3 a z - z
2
a |
In[9]:= | Kauffman[Link[11, Alternating, 475]][a, z] |
Out[9]= | 2 4 6 3 5
-2 2 4 6 a 2 a a 2 a 2 a 2 z 6 z
3 + a - a - 6 a - 4 a + -- + ---- + -- - ---- - ---- - --- - --- - 6 a z +
2 2 2 z z 3 a
z z z a
2 2 3
3 5 2 2 z 2 z 2 2 4 2 6 2 z
> 2 a z + 4 a z - 14 z + ---- - ---- - 7 a z + 9 a z + 6 a z - -- +
4 2 5
a a a
3 3 4 4
7 z 22 z 3 3 3 4 5 z 3 z 2 4
> ---- + ----- + 18 a z + 4 a z + 31 z - ---- + ---- + 22 a z -
3 a 4 2
a a a
5 5 5
4 4 6 4 z 12 z 28 z 5 3 5 5 5
> 5 a z - 4 a z + -- - ----- - ----- - 18 a z - 7 a z - 4 a z -
5 3 a
a a
6 6 7 7
6 4 z 11 z 2 6 4 6 6 6 8 z 7 z 7
> 38 z + ---- - ----- - 27 a z - 3 a z + a z + ---- + ---- - 3 a z +
4 2 3 a
a a a
8 9
5 7 8 9 z 2 8 4 8 5 z 9 3 9
> 2 a z + 17 z + ---- + 11 a z + 3 a z + ---- + 8 a z + 3 a z +
2 a
a
10 2 10
> z + a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 12 1 1 1 6 3 7 4 11
-- + 11 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- +
q 13 6 11 5 9 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
7 9 11 3 3 2 5 2 5 3
> ----- + ---- + --- + 9 q t + 10 q t + 6 q t + 9 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 L11a475 |
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