 L11a401 |
 L11a403 |
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The 3-Component Link L11a402Visit L11a402's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X10,22,5,21 X18,11,19,12 X16,9,17,10 X22,17,11,18 X8,19,9,20 X2536 X4,14,1,13 |
| Gauss Code: | {{1, -10, 2, -11}, {10, -1, 4, -9, 7, -5}, {6, -2, 11, -4, 3, -7, 8, -6, 9, -3, 5, -8}} |
| Jones Polynomial: | q-6 - 4q-5 + 9q-4 - 15q-3 + 21q-2 - 23q-1 + 25 - 20q + 16q2 - 9q3 + 4q4 - q5 |
| A2 (sl(3)) Invariant: | q-18 - q-16 + 2q-12 - 4q-10 + 4q-8 + 2q-4 + 8q-2 + 1 + 10q2 + 3q6 + 3q8 - 3q10 + 2q12 - q14 |
| HOMFLY-PT Polynomial: | a-2z-2 - 3a-2z2 - 3a-2z4 - a-2z6 - 2z-2 + 1 + 8z2 + 10z4 + 5z6 + z8 + a2z-2 - 2a2 - 8a2z2 - 7a2z4 - 2a2z6 + a4 + 2a4z2 + a4z4 |
| Kauffman Polynomial: | - a-5z3 + a-5z5 + a-4z2 - 5a-4z4 + 4a-4z6 - a-3z + 4a-3z3 - 11a-3z5 + 8a-3z7 + a-2z-2 - 9a-2z2 + 18a-2z4 - 20a-2z6 + 11a-2z8 - 2a-1z-1 - 3a-1z + 15a-1z3 - 11a-1z5 - 5a-1z7 + 8a-1z9 + 2z-2 + 3 - 31z2 + 70z4 - 64z6 + 21z8 + 2z10 - 2az-1 - 7az + 25az3 - 9az5 - 20az7 + 14az9 + a2z-2 + 4a2 - 28a2z2 + 59a2z4 - 56a2z6 + 17a2z8 + 2a2z10 - 7a3z + 21a3z3 - 19a3z5 - 3a3z7 + 6a3z9 + 2a4 - 6a4z2 + 10a4z4 - 15a4z6 + 7a4z8 - 2a5z + 6a5z3 - 9a5z5 + 4a5z7 + a6z2 - 2a6z4 + 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, 402]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 402]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 16, 21, 15], X[14, 8, 15, 7], > X[10, 22, 5, 21], X[18, 11, 19, 12], X[16, 9, 17, 10], X[22, 17, 11, 18], > X[8, 19, 9, 20], X[2, 5, 3, 6], X[4, 14, 1, 13]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {10, -1, 4, -9, 7, -5},
> {6, -2, 11, -4, 3, -7, 8, -6, 9, -3, 5, -8}] |
In[6]:= | Jones[L][q] |
Out[6]= | -6 4 9 15 21 23 2 3 4 5
25 + 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]= | -18 -16 2 4 4 2 8 2 6 8 10
1 + q - q + --- - --- + -- + -- + -- + 10 q + 3 q + 3 q - 3 q +
12 10 8 4 2
q q q q q
12 14
> 2 q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 402]][a, z] |
Out[8]= | 2 2
2 4 2 1 a 2 3 z 2 2 4 2 4
1 - 2 a + a - -- + ----- + -- + 8 z - ---- - 8 a z + 2 a z + 10 z -
2 2 2 2 2
z a z z a
4 6
3 z 2 4 4 4 6 z 2 6 8
> ---- - 7 a z + a z + 5 z - -- - 2 a z + z
2 2
a a |
In[9]:= | Kauffman[Link[11, Alternating, 402]][a, z] |
Out[9]= | 2
2 4 2 1 a 2 2 a z 3 z 3
3 + 4 a + 2 a + -- + ----- + -- - --- - --- - -- - --- - 7 a z - 7 a z -
2 2 2 2 a z z 3 a
z a z z a
2 2 3 3
5 2 z 9 z 2 2 4 2 6 2 z 4 z
> 2 a z - 31 z + -- - ---- - 28 a z - 6 a z + a z - -- + ---- +
4 2 5 3
a a a a
3 4 4
15 z 3 3 3 5 3 4 5 z 18 z 2 4
> ----- + 25 a z + 21 a z + 6 a z + 70 z - ---- + ----- + 59 a z +
a 4 2
a a
5 5 5
4 4 6 4 z 11 z 11 z 5 3 5 5 5
> 10 a z - 2 a z + -- - ----- - ----- - 9 a z - 19 a z - 9 a z -
5 3 a
a a
6 6 7 7
6 4 z 20 z 2 6 4 6 6 6 8 z 5 z
> 64 z + ---- - ----- - 56 a z - 15 a z + a z + ---- - ---- -
4 2 3 a
a a a
8 9
7 3 7 5 7 8 11 z 2 8 4 8 8 z
> 20 a z - 3 a z + 4 a z + 21 z + ----- + 17 a z + 7 a z + ---- +
2 a
a
9 3 9 10 2 10
> 14 a z + 6 a z + 2 z + 2 a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 13 1 3 1 6 3 9 6 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
10 12 11 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 L11a402 |
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