 L10a16 |
 L10a18 |
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 Knotscape |
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The 2-Component Link L10a17Visit L10a17's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X10,4,11,3 X16,8,17,7 X20,18,5,17 X18,14,19,13 X14,20,15,19 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11 |
| Gauss Code: | {{1, -9, 2, -10}, {9, -1, 3, -8, 7, -2, 10, -7, 5, -6, 8, -3, 4, -5, 6, -4}} |
| Jones Polynomial: | - q-1/2 + 2q1/2 - 6q3/2 + 9q5/2 - 13q7/2 + 13q9/2 - 13q11/2 + 11q13/2 - 7q15/2 + 4q17/2 - q19/2 |
| A2 (sl(3)) Invariant: | q-2 + q2 + 4q4 - q6 + 4q8 + 2q10 + 3q14 - 2q16 + 2q18 - 2q20 - 2q22 + q24 - 3q26 + q30 |
| HOMFLY-PT Polynomial: | - a-9z + a-7z-1 + 4a-7z + 3a-7z3 - 2a-5z-1 - 5a-5z - 5a-5z3 - 2a-5z5 - a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3 |
| Kauffman Polynomial: | a-11z3 - a-11z5 - 3a-10z2 + 7a-10z4 - 4a-10z6 + 2a-9z - 4a-9z3 + 10a-9z5 - 6a-9z7 + 2a-8 - 12a-8z2 + 17a-8z4 - 2a-8z6 - 4a-8z8 - a-7z-1 + 8a-7z - 23a-7z3 + 29a-7z5 - 12a-7z7 - a-7z9 + 5a-6 - 14a-6z2 + 10a-6z4 + 5a-6z6 - 7a-6z8 - 2a-5z-1 + 9a-5z - 21a-5z3 + 22a-5z5 - 9a-5z7 - a-5z9 + 3a-4 - 5a-4z2 + 3a-4z4 + a-4z6 - 3a-4z8 + 3a-3z5 - 3a-3z7 - a-2 + 3a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5 |
| 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[10, Alternating, 17]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, Alternating, 17]] |
Out[4]= | PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[16, 8, 17, 7], X[20, 18, 5, 17], > X[18, 14, 19, 13], X[14, 20, 15, 19], X[12, 10, 13, 9], X[8, 16, 9, 15], > X[2, 5, 3, 6], X[4, 12, 1, 11]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 7, -2, 10, -7, 5, -6, 8, -3, 4, -5,
> 6, -4}] |
In[6]:= | Jones[L][q] |
Out[6]= | 1 3/2 5/2 7/2 9/2 11/2
-(-------) + 2 Sqrt[q] - 6 q + 9 q - 13 q + 13 q - 13 q +
Sqrt[q]
13/2 15/2 17/2 19/2
> 11 q - 7 q + 4 q - q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -2 2 4 6 8 10 14 16 18 20 22
q + q + 4 q - q + 4 q + 2 q + 3 q - 2 q + 2 q - 2 q - 2 q +
24 26 30
> q - 3 q + q |
In[8]:= | HOMFLYPT[Link[10, Alternating, 17]][a, z] |
Out[8]= | 3 3 3 3 5 5 1 2 1 z 4 z 5 z 2 z 3 z 5 z z z 2 z z ---- - ---- + --- - -- + --- - --- + --- + ---- - ---- - -- + -- - ---- - -- 7 5 a z 9 7 5 a 7 5 3 a 5 3 a z a z a a a a a a a a |
In[9]:= | Kauffman[Link[10, Alternating, 17]][a, z] |
Out[9]= | 2 2
2 5 3 -2 1 2 1 2 z 8 z 9 z 3 z 3 z 12 z
-- + -- + -- - a - ---- - ---- + --- + --- + --- + --- - --- - ---- - ----- -
8 6 4 7 5 a z 9 7 5 a 10 8
a a a a z a z a a a a a
2 2 3 3 3 3 3 4 4 4
14 z 5 z z 4 z 23 z 21 z 3 z 7 z 17 z 10 z
> ----- - ---- + --- - ---- - ----- - ----- + ---- + ---- + ----- + ----- +
6 4 11 9 7 5 a 10 8 6
a a a a a a a a a
4 4 5 5 5 5 5 5 6 6
3 z 3 z z 10 z 29 z 22 z 3 z z 4 z 2 z
> ---- + ---- - --- + ----- + ----- + ----- + ---- - -- - ---- - ---- +
4 2 11 9 7 5 3 a 10 8
a a a a a a a a a
6 6 6 7 7 7 7 8 8 8 9 9
5 z z 2 z 6 z 12 z 9 z 3 z 4 z 7 z 3 z z z
> ---- + -- - ---- - ---- - ----- - ---- - ---- - ---- - ---- - ---- - -- - --
6 4 2 9 7 5 3 8 6 4 7 5
a a a a a a a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2
2 4 1 1 q 4 6 6 2 8 2 8 3
5 q + 3 q + ----- + - + -- + 6 q t + 3 q t + 7 q t + 6 q t + 6 q t +
2 2 t t
q t
10 3 10 4 12 4 12 5 14 5 14 6
> 7 q t + 7 q t + 6 q t + 4 q t + 7 q t + 3 q t +
16 6 16 7 18 7 20 8
> 4 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a17 |
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