 L11a453 |
 L11a455 |
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The 3-Component Link L11a454Visit L11a454's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,18,13,17 X16,9,17,10 X10,15,11,16 X12,20,5,19 X8,22,9,21 X2536 X4,13,1,14 |
| Gauss Code: | {{1, -10, 2, -11}, {10, -1, 4, -9, 6, -7, 3, -8}, {11, -2, 7, -6, 5, -4, 8, -3, 9, -5}} |
| Jones Polynomial: | q-6 - 3q-5 + 8q-4 - 14q-3 + 21q-2 - 23q-1 + 25 - 21q + 17q2 - 10q3 + 4q4 - q5 |
| A2 (sl(3)) Invariant: | q-20 + q-18 - 2q-16 + 2q-14 + 3q-12 - 3q-10 + 7q-8 + 3q-6 + 3q-4 + 8q-2 + 7q2 - 3q4 + 3q8 - 5q10 + 2q12 + q14 - q16 |
| HOMFLY-PT Polynomial: | - a-4z2 - a-2 + 2a-2z4 + z-2 + 4 + 2z2 - z6 - 2a2z-2 - 3a2 + a2z2 + 3a2z4 + a4z-2 - a4 - 3a4z2 + a6 |
| Kauffman Polynomial: | - a-5z3 + a-5z5 + a-4z2 - 4a-4z4 + 4a-4z6 - 2a-3z + 8a-3z3 - 13a-3z5 + 9a-3z7 + 3a-2 - 10a-2z2 + 16a-2z4 - 20a-2z6 + 12a-2z8 - 7a-1z + 19a-1z3 - 19a-1z5 - a-1z7 + 8a-1z9 - z-2 + 11 - 32z2 + 53z4 - 53z6 + 20z8 + 2z10 + 2az-1 - 12az + 19az3 - 8az5 - 16az7 + 13az9 - 2a2z-2 + 11a2 - 27a2z2 + 41a2z4 - 39a2z6 + 13a2z8 + 2a2z10 + 2a3z-1 - 8a3z + 14a3z3 - 10a3z5 - 3a3z7 + 5a3z9 - a4z-2 + 3a4 - 3a4z2 + 5a4z4 - 9a4z6 + 5a4z8 - a5z + 5a5z3 - 7a5z5 + 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, 454]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 454]] |
Out[4]= | PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[20, 12, 21, 11], X[18, 8, 19, 7], > X[22, 18, 13, 17], X[16, 9, 17, 10], X[10, 15, 11, 16], X[12, 20, 5, 19], > X[8, 22, 9, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 6, -7, 3, -8},
> {11, -2, 7, -6, 5, -4, 8, -3, 9, -5}] |
In[6]:= | Jones[L][q] |
Out[6]= | -6 3 8 14 21 23 2 3 4 5
25 + q - -- + -- - -- + -- - -- - 21 q + 17 q - 10 q + 4 q - q
5 4 3 2 q
q q q q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -20 -18 2 2 3 3 7 3 3 8 2 4 8
q + q - --- + --- + --- - --- + -- + -- + -- + -- + 7 q - 3 q + 3 q -
16 14 12 10 8 6 4 2
q q q q q q q q
10 12 14 16
> 5 q + 2 q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 454]][a, z] |
Out[8]= | 2 4 2
-2 2 4 6 -2 2 a a 2 z 2 2 4 2
4 - a - 3 a - a + a + z - ---- + -- + 2 z - -- + a z - 3 a z +
2 2 4
z z a
4
2 z 2 4 6
> ---- + 3 a z - z
2
a |
In[9]:= | Kauffman[Link[11, Alternating, 454]][a, z] |
Out[9]= | 2 4 3
3 2 4 6 -2 2 a a 2 a 2 a 2 z 7 z
11 + -- + 11 a + 3 a - a - z - ---- - -- + --- + ---- - --- - --- -
2 2 2 z z 3 a
a z z a
2 2
3 5 2 z 10 z 2 2 4 2
> 12 a z - 8 a z - a z - 32 z + -- - ----- - 27 a z - 3 a z +
4 2
a a
3 3 3 4
6 2 z 8 z 19 z 3 3 3 5 3 4 4 z
> 3 a z - -- + ---- + ----- + 19 a z + 14 a z + 5 a z + 53 z - ---- +
5 3 a 4
a a a
4 5 5 5
16 z 2 4 4 4 6 4 z 13 z 19 z 5
> ----- + 41 a z + 5 a z - 3 a z + -- - ----- - ----- - 8 a z -
2 5 3 a
a a a
6 6
3 5 5 5 6 4 z 20 z 2 6 4 6 6 6
> 10 a z - 7 a z - 53 z + ---- - ----- - 39 a z - 9 a z + a z +
4 2
a a
7 7 8
9 z z 7 3 7 5 7 8 12 z 2 8
> ---- - -- - 16 a z - 3 a z + 3 a z + 20 z + ----- + 13 a z +
3 a 2
a a
9
4 8 8 z 9 3 9 10 2 10
> 5 a z + ---- + 13 a z + 5 a z + 2 z + 2 a z
a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 14 1 2 1 6 3 9 5 12
-- + 14 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
9 11 12 3 3 2 5 2 5 3
> ----- + ---- + --- + 10 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
> 7 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a454 |
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