 L11a4 |
 L11a6 |
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 Knotscape |
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The 2-Component Link L11a5Visit L11a5's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,12,5,11 X20,14,21,13 X14,20,15,19 X12,22,13,21 X18,10,19,9 X2,16,3,15 |
| Gauss Code: | {{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 7, -8, 11, -2, 3, -10, 8, -7, 9, -6}} |
| Jones Polynomial: | q-5/2 - 4q-3/2 + 6q-1/2 - 10q1/2 + 12q3/2 - 14q5/2 + 13q7/2 - 11q9/2 + 8q11/2 - 5q13/2 + 3q15/2 - q17/2 |
| A2 (sl(3)) Invariant: | - q-8 + 2q-6 + q-4 + 5 + q4 + q6 - q8 + 2q10 - q12 + q14 + q16 - 3q18 + q20 - q24 + q26 |
| HOMFLY-PT Polynomial: | - a-7z - a-7z3 + 2a-5z + 2a-5z3 + a-5z5 - a-3z + a-3z5 - a-1z-1 + a-1z3 + a-1z5 + az-1 - az3 |
| Kauffman Polynomial: | - 3a-9z3 + 4a-9z5 - a-9z7 + 6a-8z2 - 15a-8z4 + 13a-8z6 - 3a-8z8 - 2a-7z + 11a-7z3 - 21a-7z5 + 17a-7z7 - 4a-7z9 + 11a-6z2 - 27a-6z4 + 16a-6z6 + 2a-6z8 - 2a-6z10 - 4a-5z + 22a-5z3 - 46a-5z5 + 36a-5z7 - 9a-5z9 + 6a-4z2 - 23a-4z4 + 18a-4z6 - a-4z8 - 2a-4z10 - 2a-3z + 8a-3z3 - 13a-3z5 + 12a-3z7 - 5a-3z9 + a-2z2 - 4a-2z4 + 9a-2z6 - 6a-2z8 - a-1z-1 + 4a-1z3 + 4a-1z5 - 6a-1z7 + 1 + 6z4 - 6z6 - az-1 + 4az3 - 4az5 - a2z4 |
| 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]= | 2 |
In[3]:= | Show[DrawMorseLink[Link[11, Alternating, 5]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 5]] |
Out[4]= | PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[10, 6, 11, 5], > X[8, 4, 9, 3], X[22, 12, 5, 11], X[20, 14, 21, 13], X[14, 20, 15, 19], > X[12, 22, 13, 21], X[18, 10, 19, 9], X[2, 16, 3, 15]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 7, -8, 11, -2, 3, -10,
> 8, -7, 9, -6}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(5/2) 4 6 3/2 5/2 7/2 9/2
q - ---- + ------- - 10 Sqrt[q] + 12 q - 14 q + 13 q - 11 q +
3/2 Sqrt[q]
q
11/2 13/2 15/2 17/2
> 8 q - 5 q + 3 q - q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -8 2 -4 4 6 8 10 12 14 16 18 20
5 - q + -- + q + q + q - q + 2 q - q + q + q - 3 q + q -
6
q
24 26
> q + q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 5]][a, z] |
Out[8]= | 3 3 3 5 5 5
1 a z 2 z z z 2 z z 3 z z z
-(---) + - - -- + --- - -- - -- + ---- + -- - a z + -- + -- + --
a z z 7 5 3 7 5 a 5 3 a
a a a a a a a |
In[9]:= | Kauffman[Link[11, Alternating, 5]][a, z] |
Out[9]= | 2 2 2 2 3 3
1 a 2 z 4 z 2 z 6 z 11 z 6 z z 3 z 11 z
1 - --- - - - --- - --- - --- + ---- + ----- + ---- + -- - ---- + ----- +
a z z 7 5 3 8 6 4 2 9 7
a a a a a a a a a
3 3 3 4 4 4 4
22 z 8 z 4 z 3 4 15 z 27 z 23 z 4 z
> ----- + ---- + ---- + 4 a z + 6 z - ----- - ----- - ----- - ---- -
5 3 a 8 6 4 2
a a a a a a
5 5 5 5 5 6
2 4 4 z 21 z 46 z 13 z 4 z 5 6 13 z
> a z + ---- - ----- - ----- - ----- + ---- - 4 a z - 6 z + ----- +
9 7 5 3 a 8
a a a a a
6 6 6 7 7 7 7 7 8 8
16 z 18 z 9 z z 17 z 36 z 12 z 6 z 3 z 2 z
> ----- + ----- + ---- - -- + ----- + ----- + ----- - ---- - ---- + ---- -
6 4 2 9 7 5 3 a 8 6
a a a a a a a a a
8 8 9 9 9 10 10
z 6 z 4 z 9 z 5 z 2 z 2 z
> -- - ---- - ---- - ---- - ---- - ----- - -----
4 2 7 5 3 6 4
a a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 1 3 1 3 3 2 4 4 2
7 + 5 q + ----- + ----- + ----- + - + ---- + 7 q t + 5 q t + 7 q t +
6 3 4 2 2 2 t 2
q t q t q t q t
6 2 6 3 8 3 8 4 10 4 10 5 12 5
> 7 q t + 6 q t + 7 q t + 5 q t + 6 q t + 3 q t + 5 q t +
12 6 14 6 14 7 16 7 18 8
> 2 q t + 3 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a5 |
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