 L10a21 |
 L10a23 |
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 Knotscape |
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The 2-Component Link L10a22Visit L10a22's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X14,9,15,10 X4,19,1,20 X12,6,13,5 X10,4,11,3 X20,12,5,11 X2,14,3,13 X18,16,19,15 X16,8,17,7 X8,18,9,17 |
| Gauss Code: | {{1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8, 3, -6}} |
| Jones Polynomial: | q-3/2 - 4q-1/2 + 7q1/2 - 12q3/2 + 14q5/2 - 17q7/2 + 15q9/2 - 12q11/2 + 9q13/2 - 4q15/2 + q17/2 |
| A2 (sl(3)) Invariant: | - q-4 + 2q-2 - 1 + 2q2 + 4q4 - q6 + 6q8 + 3q12 + q14 - 3q16 + q18 - 4q20 + q24 - q26 |
| HOMFLY-PT Polynomial: | a-7z-1 + a-7z + a-7z3 - 2a-5z-1 - 4a-5z - 5a-5z3 - 2a-5z5 + 3a-3z + 6a-3z3 + 4a-3z5 + a-3z7 + a-1z-1 - 2a-1z3 - a-1z5 |
| Kauffman Polynomial: | - a-10z4 + a-9z3 - 4a-9z5 + 2a-8 - 5a-8z2 + 10a-8z4 - 9a-8z6 - a-7z-1 + 2a-7z - 8a-7z3 + 15a-7z5 - 11a-7z7 + 5a-6 - 14a-6z2 + 19a-6z4 - a-6z6 - 7a-6z8 - 2a-5z-1 + 7a-5z - 26a-5z3 + 41a-5z5 - 16a-5z7 - 2a-5z9 + 3a-4 - 11a-4z2 + 4a-4z4 + 19a-4z6 - 12a-4z8 + 6a-3z - 26a-3z3 + 33a-3z5 - 9a-3z7 - 2a-3z9 - a-2 - 3a-2z2 - 2a-2z4 + 10a-2z6 - 5a-2z8 + a-1z-1 + a-1z - 9a-1z3 + 11a-1z5 - 4a-1z7 - z2 + 2z4 - z6 |
| 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, 22]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, Alternating, 22]] |
Out[4]= | PD[X[6, 1, 7, 2], X[14, 9, 15, 10], X[4, 19, 1, 20], X[12, 6, 13, 5], > X[10, 4, 11, 3], X[20, 12, 5, 11], X[2, 14, 3, 13], X[18, 16, 19, 15], > X[16, 8, 17, 7], X[8, 18, 9, 17]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8,
> 3, -6}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(3/2) 4 3/2 5/2 7/2 9/2
q - ------- + 7 Sqrt[q] - 12 q + 14 q - 17 q + 15 q -
Sqrt[q]
11/2 13/2 15/2 17/2
> 12 q + 9 q - 4 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -4 2 2 4 6 8 12 14 16 18 20
-1 - q + -- + 2 q + 4 q - q + 6 q + 3 q + q - 3 q + q - 4 q +
2
q
24 26
> q - q |
In[8]:= | HOMFLYPT[Link[10, Alternating, 22]][a, z] |
Out[8]= | 3 3 3 3 5 5
1 2 1 z 4 z 3 z z 5 z 6 z 2 z 2 z 4 z
---- - ---- + --- + -- - --- + --- + -- - ---- + ---- - ---- - ---- + ---- -
7 5 a z 7 5 3 7 5 3 a 5 3
a z a z a a a a a a a a
5 7
z z
> -- + --
a 3
a |
In[9]:= | Kauffman[Link[10, Alternating, 22]][a, z] |
Out[9]= | 2
2 5 3 -2 1 2 1 2 z 7 z 6 z z 2 5 z
-- + -- + -- - a - ---- - ---- + --- + --- + --- + --- + - - z - ---- -
8 6 4 7 5 a z 7 5 3 a 8
a a a a z a z a a a a
2 2 2 3 3 3 3 3 4
14 z 11 z 3 z z 8 z 26 z 26 z 9 z 4 z
> ----- - ----- - ---- + -- - ---- - ----- - ----- - ---- + 2 z - --- +
6 4 2 9 7 5 3 a 10
a a a a a a a a
4 4 4 4 5 5 5 5 5
10 z 19 z 4 z 2 z 4 z 15 z 41 z 33 z 11 z 6
> ----- + ----- + ---- - ---- - ---- + ----- + ----- + ----- + ----- - z -
8 6 4 2 9 7 5 3 a
a a a a a a a a
6 6 6 6 7 7 7 7 8 8
9 z z 19 z 10 z 11 z 16 z 9 z 4 z 7 z 12 z
> ---- - -- + ----- + ----- - ----- - ----- - ---- - ---- - ---- - ----- -
8 6 4 2 7 5 3 a 6 4
a a a a a a a a a
8 9 9
5 z 2 z 2 z
> ---- - ---- - ----
2 5 3
a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2
2 4 1 -2 3 4 3 q 4 6 6 2
8 q + 6 q + ----- + t + ----- + - + ---- + 8 q t + 6 q t + 9 q t +
4 3 2 2 t t
q t q t
8 2 8 3 10 3 10 4 12 4 12 5 14 5
> 8 q t + 6 q t + 9 q t + 6 q t + 6 q t + 3 q t + 6 q t +
14 6 16 6 18 7
> q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a22 |
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