 L10a60 |
 L10a62 |
| © | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: |
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 Knotscape |
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The 2-Component Link L10a61Visit L10a61's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X10,4,11,3 X20,10,7,9 X2738 X18,14,19,13 X6,12,1,11 X16,20,17,19 X4,16,5,15 X14,6,15,5 X12,18,13,17 |
| Gauss Code: | {{1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -10, 5, -9, 8, -7, 10, -5, 7, -3}} |
| Jones Polynomial: | - q-1/2 + 3q1/2 - 7q3/2 + 10q5/2 - 14q7/2 + 14q9/2 - 14q11/2 + 11q13/2 - 7q15/2 + 4q17/2 - q19/2 |
| A2 (sl(3)) Invariant: | q-2 - 1 + 3q4 - 2q6 + 4q8 + 2q10 + q12 + 4q14 - q16 + 3q18 - 2q20 - 2q22 + q24 - 3q26 + q30 |
| HOMFLY-PT Polynomial: | - a-9z + a-7z-1 + 4a-7z + 3a-7z3 - 3a-5z-1 - 6a-5z - 5a-5z3 - 2a-5z5 + 2a-3z-1 + 3a-3z - a-3z5 + a-1z + a-1z3 |
| Kauffman Polynomial: | a-11z3 - a-11z5 - 3a-10z2 + 7a-10z4 - 4a-10z6 + 2a-9z - 5a-9z3 + 10a-9z5 - 6a-9z7 + a-8 - 12a-8z2 + 18a-8z4 - 3a-8z6 - 4a-8z8 - a-7z-1 + 10a-7z - 28a-7z3 + 34a-7z5 - 14a-7z7 - a-7z9 + 3a-6 - 14a-6z2 + 13a-6z4 + 5a-6z6 - 8a-6z8 - 3a-5z-1 + 15a-5z - 32a-5z3 + 33a-5z5 - 13a-5z7 - a-5z9 + 3a-4 - 7a-4z2 + 7a-4z4 + a-4z6 - 4a-4z8 - 2a-3z-1 + 6a-3z - 8a-3z3 + 9a-3z5 - 5a-3z7 - 2a-2z2 + 5a-2z4 - 3a-2z6 - a-1z + 2a-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, 61]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[10, Alternating, 61]] |
Out[4]= | PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[18, 14, 19, 13], X[6, 12, 1, 11], X[16, 20, 17, 19], X[4, 16, 5, 15], > X[14, 6, 15, 5], X[12, 18, 13, 17]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -4, 2, -8, 9, -6},
> {4, -1, 3, -2, 6, -10, 5, -9, 8, -7, 10, -5, 7, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | 1 3/2 5/2 7/2 9/2 11/2
-(-------) + 3 Sqrt[q] - 7 q + 10 q - 14 q + 14 q - 14 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 4 6 8 10 12 14 16 18 20
-1 + q + 3 q - 2 q + 4 q + 2 q + q + 4 q - q + 3 q - 2 q -
22 24 26 30
> 2 q + q - 3 q + q |
In[8]:= | HOMFLYPT[Link[10, Alternating, 61]][a, z] |
Out[8]= | 3 3 3 5 5 1 3 2 z 4 z 6 z 3 z z 3 z 5 z z 2 z z ---- - ---- + ---- - -- + --- - --- + --- + - + ---- - ---- + -- - ---- - -- 7 5 3 9 7 5 3 a 7 5 a 5 3 a z a z a z a a a a a a a a |
In[9]:= | Kauffman[Link[10, Alternating, 61]][a, z] |
Out[9]= | 2
-8 3 3 1 3 2 2 z 10 z 15 z 6 z z 3 z
a + -- + -- - ---- - ---- - ---- + --- + ---- + ---- + --- - - - ---- -
6 4 7 5 3 9 7 5 3 a 10
a a a z a z a z a a a a a
2 2 2 2 3 3 3 3 3 3
12 z 14 z 7 z 2 z z 5 z 28 z 32 z 8 z 2 z
> ----- - ----- - ---- - ---- + --- - ---- - ----- - ----- - ---- + ---- +
8 6 4 2 11 9 7 5 3 a
a a a a a a a a a
4 4 4 4 4 5 5 5 5 5
7 z 18 z 13 z 7 z 5 z z 10 z 34 z 33 z 9 z
> ---- + ----- + ----- + ---- + ---- - --- + ----- + ----- + ----- + ---- -
10 8 6 4 2 11 9 7 5 3
a a a a a a a a a a
5 6 6 6 6 6 7 7 7 7 8
z 4 z 3 z 5 z z 3 z 6 z 14 z 13 z 5 z 4 z
> -- - ---- - ---- + ---- + -- - ---- - ---- - ----- - ----- - ---- - ---- -
a 10 8 6 4 2 9 7 5 3 8
a a a a a a a a a a
8 8 9 9
8 z 4 z z z
> ---- - ---- - -- - --
6 4 7 5
a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2
2 4 1 2 q 4 6 6 2 8 2 8 3
5 q + 3 q + ----- + - + -- + 6 q t + 4 q t + 8 q t + 7 q t + 7 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 + 7 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 L10a61 |
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