 L11n309 |
 L11n311 |
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The 3-Component Link L11n310Visit L11n310's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X3,13,4,12 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,20,10,21 X15,5,16,10 X19,15,20,14 X2536 X11,1,12,4 |
| Gauss Code: | {{1, -10, -2, 11}, {10, -1, -5, 6, -7, 8}, {-11, 2, -3, 9, -8, 5, -4, 3, -9, 7, -6, 4}} |
| Jones Polynomial: | 3q-1 - 5 + 10q - 11q2 + 14q3 - 12q4 + 10q5 - 7q6 + 3q7 - q8 |
| A2 (sl(3)) Invariant: | 3q-4 + 2q-2 + 3 + 8q2 + 3q4 + 7q6 + 4q8 + 2q10 + 3q12 - 3q14 + q16 - 3q18 - 3q20 + q22 - q24 |
| HOMFLY-PT Polynomial: | - a-6z-2 - 3a-6 - 2a-6z2 - a-6z4 + 4a-4z-2 + 11a-4 + 10a-4z2 + 4a-4z4 + a-4z6 - 5a-2z-2 - 13a-2 - 12a-2z2 - 4a-2z4 + 2z-2 + 5 + 3z2 |
| Kauffman Polynomial: | a-9z - 2a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 3a-8z6 + a-7z-1 - 4a-7z + 5a-7z3 - 9a-7z5 + 5a-7z7 - a-6z-2 + 5a-6 - 8a-6z2 + 8a-6z4 - 9a-6z6 + 5a-6z8 + 5a-5z-1 - 21a-5z + 32a-5z3 - 22a-5z5 + 5a-5z7 + 2a-5z9 - 4a-4z-2 + 18a-4 - 36a-4z2 + 44a-4z4 - 29a-4z6 + 10a-4z8 + 9a-3z-1 - 29a-3z + 35a-3z3 - 18a-3z5 + 3a-3z7 + 2a-3z9 - 5a-2z-2 + 21a-2 - 41a-2z2 + 37a-2z4 - 17a-2z6 + 5a-2z8 + 5a-1z-1 - 13a-1z + 10a-1z3 - 6a-1z5 + 3a-1z7 - 2z-2 + 9 - 14z2 + 6z4 |
| 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, 310]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 310]] |
Out[4]= | PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[13, 19, 14, 18], X[17, 11, 18, 22], > X[7, 17, 8, 16], X[21, 8, 22, 9], X[9, 20, 10, 21], X[15, 5, 16, 10], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[11, 1, 12, 4]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, -2, 11}, {10, -1, -5, 6, -7, 8},
> {-11, 2, -3, 9, -8, 5, -4, 3, -9, 7, -6, 4}] |
In[6]:= | Jones[L][q] |
Out[6]= | 3 2 3 4 5 6 7 8
-5 + - + 10 q - 11 q + 14 q - 12 q + 10 q - 7 q + 3 q - q
q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 3 2 2 4 6 8 10 12 14 16 18
3 + -- + -- + 8 q + 3 q + 7 q + 4 q + 2 q + 3 q - 3 q + q - 3 q -
4 2
q q
20 22 24
> 3 q + q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 310]][a, z] |
Out[8]= | 2 2 2
3 11 13 2 1 4 5 2 2 z 10 z 12 z
5 - -- + -- - -- + -- - ----- + ----- - ----- + 3 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 4 z z
> -- + ---- - ---- + --
6 4 2 4
a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 310]][a, z] |
Out[9]= | 5 18 21 2 1 4 5 1 5 9 5 z
9 + -- + -- + -- - -- - ----- - ----- - ----- + ---- + ---- + ---- + --- + -- -
6 4 2 2 6 2 4 2 2 2 7 5 3 a z 9
a a a z a z a z a z a z a z a z a
2 2 2 2 3
4 z 21 z 29 z 13 z 2 z 8 z 36 z 41 z 2 z
> --- - ---- - ---- - ---- - 14 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
5 z 32 z 35 z 10 z 4 5 z 8 z 44 z 37 z z
> ---- + ----- + ----- + ----- + 6 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
9 z 22 z 18 z 6 z 3 z 9 z 29 z 17 z 5 z 5 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
3 z 3 z 5 z 10 z 5 z 2 z 2 z
> ---- + ---- + ---- + ----- + ---- + ---- + ----
3 a 6 4 2 5 3
a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3 3 1 3 2 q 3 5 5 2 7 2
7 q + 5 q + ----- + ---- + --- + --- + 6 q t + 5 q t + 8 q t + 7 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
> 5 q t + 7 q t + 5 q t + 5 q t + 2 q t + 5 q t + q t +
15 6 17 7
> 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n310 |
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