 L11n294 |
 L11n296 |
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The 3-Component Link L11n295Visit L11n295's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X21,10,22,5 X11,19,12,18 X9,17,10,16 X17,11,18,22 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: | 2q-1 - 3 + 8q - 8q2 + 11q3 - 10q4 + 8q5 - 6q6 + 3q7 - q8 |
| A2 (sl(3)) Invariant: | 2q-4 + 2q-2 + 3 + 8q2 + 5q4 + 8q6 + 5q8 + q10 + q12 - 4q14 - 2q18 - 2q20 + q22 - q24 |
| HOMFLY-PT Polynomial: | - a-6z-2 - 2a-6 - 2a-6z2 - a-6z4 + 4a-4z-2 + 8a-4 + 8a-4z2 + 4a-4z4 + a-4z6 - 5a-2z-2 - 10a-2 - 9a-2z2 - 3a-2z4 + 2z-2 + 4 + 2z2 |
| Kauffman Polynomial: | - 2a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 3a-8z6 + a-7z-1 - 5a-7z + 11a-7z3 - 13a-7z5 + 5a-7z7 - a-6z-2 + 3a-6 - 6a-6z2 + 9a-6z4 - 9a-6z6 + 4a-6z8 + 5a-5z-1 - 18a-5z + 32a-5z3 - 23a-5z5 + 6a-5z7 + a-5z9 - 4a-4z-2 + 12a-4 - 24a-4z2 + 30a-4z4 - 18a-4z6 + 6a-4z8 + 9a-3z-1 - 24a-3z + 24a-3z3 - 10a-3z5 + 2a-3z7 + a-3z9 - 5a-2z-2 + 15a-2 - 25a-2z2 + 18a-2z4 - 6a-2z6 + 2a-2z8 + 5a-1z-1 - 11a-1z + 5a-1z3 - a-1z5 + a-1z7 - 2z-2 + 7 - 8z2 + 3z4 |
| 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, NonAlternating, 295]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 295]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 16, 21, 15], X[14, 8, 15, 7], > X[21, 10, 22, 5], X[11, 19, 12, 18], X[9, 17, 10, 16], X[17, 11, 18, 22], > 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]= | 2 2 3 4 5 6 7 8
-3 + - + 8 q - 8 q + 11 q - 10 q + 8 q - 6 q + 3 q - q
q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 2 2 2 4 6 8 10 12 14 18 20
3 + -- + -- + 8 q + 5 q + 8 q + 5 q + q + q - 4 q - 2 q - 2 q +
4 2
q q
22 24
> q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 295]][a, z] |
Out[8]= | 2 2 2
2 8 10 2 1 4 5 2 2 z 8 z 9 z
4 - -- + -- - -- + -- - ----- + ----- - ----- + 2 z - ---- + ---- - ---- -
6 4 2 2 6 2 4 2 2 2 6 4 2
a a a z a z a z a z a a a
4 4 4 6
z 4 z 3 z z
> -- + ---- - ---- + --
6 4 2 4
a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 295]][a, z] |
Out[9]= | 3 12 15 2 1 4 5 1 5 9 5
7 + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- -
6 4 2 2 6 2 4 2 2 2 7 5 3 a z
a a a z a z a z a z a z a z a z
2 2 2 2 3
5 z 18 z 24 z 11 z 2 z 6 z 24 z 25 z 2 z
> --- - ---- - ---- - ---- - 8 z + -- - ---- - ----- - ----- - ---- +
7 5 3 a 8 6 4 2 9
a a a a a a a a
3 3 3 3 4 4 4 4 5
11 z 32 z 24 z 5 z 4 6 z 9 z 30 z 18 z z
> ----- + ----- + ----- + ---- + 3 z - ---- + ---- + ----- + ----- + -- -
7 5 3 a 8 6 4 2 9
a a a a a a a a
5 5 5 5 6 6 6 6 7 7
13 z 23 z 10 z z 3 z 9 z 18 z 6 z 5 z 6 z
> ----- - ----- - ----- - -- + ---- - ---- - ----- - ---- + ---- + ---- +
7 5 3 a 8 6 4 2 7 5
a a a a a a a a a
7 7 8 8 8 9 9
2 z z 4 z 6 z 2 z z z
> ---- + -- + ---- + ---- + ---- + -- + --
3 a 6 4 2 5 3
a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3 2 1 2 q 3 5 5 2 7 2
6 q + 4 q + ----- + ---- + --- + - + 4 q t + 4 q t + 7 q t + 5 q t +
3 2 2 q t t
q t q t
7 3 9 3 9 4 11 4 11 5 13 5 13 6
> 4 q t + 6 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t +
15 6 17 7
> 2 q t + q t |
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