 L11a370 |
 L11a372 |
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 Knotscape |
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The 2-Component Link L11a371Visit L11a371's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X12,1,13,2 X14,4,15,3 X22,14,11,13 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,7,21 X20,8,21,7 X18,10,19,9 X10,18,1,17 X8,20,9,19 |
| Gauss Code: | {{1, -5, 2, -6, 4, -7, 8, -11, 9, -10}, {5, -1, 3, -2, 6, -4, 10, -9, 11, -8, 7, -3}} |
| Jones Polynomial: | - q-1/2 + 2q1/2 - 4q3/2 + 6q5/2 - 8q7/2 + 9q9/2 - 9q11/2 + 7q13/2 - 6q15/2 + 3q17/2 - 2q19/2 + q21/2 |
| A2 (sl(3)) Invariant: | q-2 + q4 - 2q6 + q8 + 2q14 + 2q18 + q20 + q22 + 3q24 - q32 |
| HOMFLY-PT Polynomial: | 2a-9z + a-9z3 - a-7z-1 - 3a-7z - 3a-7z3 - a-7z5 + a-5z-1 + 2a-5z - a-5z3 - a-5z5 - 2a-3z3 - a-3z5 + 2a-1z + a-1z3 |
| Kauffman Polynomial: | - 3a-12z2 + 4a-12z4 - a-12z6 + 3a-11z - 8a-11z3 + 8a-11z5 - 2a-11z7 - a-10z2 - 2a-10z4 + 6a-10z6 - 2a-10z8 + 3a-9z + a-9z3 - 7a-9z5 + 7a-9z7 - 2a-9z9 + 2a-8z2 - 5a-8z4 + a-8z6 + 2a-8z8 - a-8z10 + a-7z-1 - 8a-7z + 23a-7z3 - 33a-7z5 + 18a-7z7 - 4a-7z9 - a-6 + 5a-6z2 - 8a-6z4 + 2a-6z8 - a-6z10 + a-5z-1 - 6a-5z + 12a-5z3 - 14a-5z5 + 7a-5z7 - 2a-5z9 + 3a-4z2 - 4a-4z4 + 4a-4z6 - 2a-4z8 + a-3z3 + 3a-3z5 - 2a-3z7 - 2a-2z2 + 5a-2z4 - 2a-2z6 - 2a-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[11, Alternating, 371]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 371]] |
Out[4]= | PD[X[12, 1, 13, 2], X[14, 4, 15, 3], X[22, 14, 11, 13], X[16, 6, 17, 5], > X[2, 11, 3, 12], X[4, 16, 5, 15], X[6, 22, 7, 21], X[20, 8, 21, 7], > X[18, 10, 19, 9], X[10, 18, 1, 17], X[8, 20, 9, 19]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -5, 2, -6, 4, -7, 8, -11, 9, -10},
> {5, -1, 3, -2, 6, -4, 10, -9, 11, -8, 7, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | 1 3/2 5/2 7/2 9/2 11/2
-(-------) + 2 Sqrt[q] - 4 q + 6 q - 8 q + 9 q - 9 q +
Sqrt[q]
13/2 15/2 17/2 19/2 21/2
> 7 q - 6 q + 3 q - 2 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -2 4 6 8 14 18 20 22 24 32 q + q - 2 q + q + 2 q + 2 q + q + q + 3 q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 371]][a, z] |
Out[8]= | 3 3 3 3 3 5 5
1 1 2 z 3 z 2 z 2 z z 3 z z 2 z z z z
-(----) + ---- + --- - --- + --- + --- + -- - ---- - -- - ---- + -- - -- - -- -
7 5 9 7 5 a 9 7 5 3 a 7 5
a z a z a a a a a a a a a
5
z
> --
3
a |
In[9]:= | Kauffman[Link[11, Alternating, 371]][a, z] |
Out[9]= | 2 2 2 2
-6 1 1 3 z 3 z 8 z 6 z 2 z 3 z z 2 z 5 z
-a + ---- + ---- + --- + --- - --- - --- - --- - ---- - --- + ---- + ---- +
7 5 11 9 7 5 a 12 10 8 6
a z a z a a a a a a a a
2 2 3 3 3 3 3 3 4 4 4
3 z 2 z 8 z z 23 z 12 z z 3 z 4 z 2 z 5 z
> ---- - ---- - ---- + -- + ----- + ----- + -- + ---- + ---- - ---- - ---- -
4 2 11 9 7 5 3 a 12 10 8
a a a a a a a a a a
4 4 4 5 5 5 5 5 5 6 6
8 z 4 z 5 z 8 z 7 z 33 z 14 z 3 z z z 6 z
> ---- - ---- + ---- + ---- - ---- - ----- - ----- + ---- - -- - --- + ---- +
6 4 2 11 9 7 5 3 a 12 10
a a a a a a a a a a
6 6 6 7 7 7 7 7 8 8 8
z 4 z 2 z 2 z 7 z 18 z 7 z 2 z 2 z 2 z 2 z
> -- + ---- - ---- - ---- + ---- + ----- + ---- - ---- - ---- + ---- + ---- -
8 4 2 11 9 7 5 3 10 8 6
a a a a a a a a a a a
8 9 9 9 10 10
2 z 2 z 4 z 2 z z z
> ---- - ---- - ---- - ---- - --- - ---
4 9 7 5 8 6
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
3 q + 2 q + ----- + - + -- + 4 q t + 2 q t + 4 q t + 4 q t + 5 q t +
2 2 t t
q t
10 3 10 4 12 4 12 5 14 5 14 6
> 4 q t + 4 q t + 5 q t + 3 q t + 4 q t + 3 q t +
16 6 16 7 18 7 18 8 20 8 22 9
> 4 q t + q t + 2 q t + q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a371 |
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