 L11a221 |
 L11a223 |
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The 2-Component Link L11a222Visit L11a222's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X8192 X12,4,13,3 X22,10,7,9 X20,12,21,11 X10,22,11,21 X16,6,17,5 X18,16,19,15 X14,20,15,19 X2738 X4,14,5,13 X6,18,1,17 |
| Gauss Code: | {{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -4, 5, -3}} |
| Jones Polynomial: | - q-1/2 + 2q1/2 - 6q3/2 + 9q5/2 - 13q7/2 + 15q9/2 - 15q11/2 + 13q13/2 - 10q15/2 + 6q17/2 - 3q19/2 + q21/2 |
| A2 (sl(3)) Invariant: | q-2 + q2 + 4q4 - q6 + 3q8 + q10 - 2q12 + 2q14 - 2q16 + 2q18 - q22 + 3q24 - 2q26 + q30 - q32 |
| HOMFLY-PT Polynomial: | a-9z + a-9z3 - a-7z3 - a-7z5 - 2a-5z - 4a-5z3 - 2a-5z5 - a-3z-1 - a-3z3 - a-3z5 + a-1z-1 + 2a-1z + a-1z3 |
| Kauffman Polynomial: | - 2a-12z2 + 3a-12z4 - a-12z6 + a-11z - 7a-11z3 + 9a-11z5 - 3a-11z7 + a-10z2 - 6a-10z4 + 10a-10z6 - 4a-10z8 + a-9z5 + 4a-9z7 - 3a-9z9 - a-8z2 - 2a-8z4 + 9a-8z6 - 4a-8z8 - a-8z10 + 2a-7z3 - 4a-7z5 + 7a-7z7 - 5a-7z9 - 5a-6z2 + 6a-6z4 + a-6z6 - 3a-6z8 - a-6z10 + 2a-5z - 7a-5z3 + 8a-5z5 - 3a-5z7 - 2a-5z9 - a-4z2 + 2a-4z4 + a-4z6 - 3a-4z8 + a-3z-1 - 2a-3z + a-3z3 + 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[11, Alternating, 222]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 222]] |
Out[4]= | PD[X[8, 1, 9, 2], X[12, 4, 13, 3], X[22, 10, 7, 9], X[20, 12, 21, 11], > X[10, 22, 11, 21], X[16, 6, 17, 5], X[18, 16, 19, 15], X[14, 20, 15, 19], > X[2, 7, 3, 8], X[4, 14, 5, 13], X[6, 18, 1, 17]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -9, 2, -10, 6, -11},
> {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -4, 5, -3}] |
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 + 15 q - 15 q +
Sqrt[q]
13/2 15/2 17/2 19/2 21/2
> 13 q - 10 q + 6 q - 3 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -2 2 4 6 8 10 12 14 16 18 22
q + q + 4 q - q + 3 q + q - 2 q + 2 q - 2 q + 2 q - q +
24 26 30 32
> 3 q - 2 q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 222]][a, z] |
Out[8]= | 3 3 3 3 3 5 5 5 1 1 z 2 z 2 z z z 4 z z z z 2 z z -(----) + --- + -- - --- + --- + -- - -- - ---- - -- + -- - -- - ---- - -- 3 a z 9 5 a 9 7 5 3 a 7 5 3 a z a a a a a a a a a |
In[9]:= | Kauffman[Link[11, Alternating, 222]][a, z] |
Out[9]= | 2 2 2 2 2
-2 1 1 z 2 z 2 z 3 z 2 z z z 5 z z
-a + ---- + --- + --- + --- - --- - --- - ---- + --- - -- - ---- - -- -
3 a z 11 5 3 a 12 10 8 6 4
a z a a a a a a a a
3 3 3 3 3 4 4 4 4 4 4
7 z 2 z 7 z z 3 z 3 z 6 z 2 z 6 z 2 z 3 z
> ---- + ---- - ---- + -- + ---- + ---- - ---- - ---- + ---- + ---- + ---- +
11 7 5 3 a 12 10 8 6 4 2
a a a a a a a a a a
5 5 5 5 5 5 6 6 6 6 6 6
9 z z 4 z 8 z 3 z z z 10 z 9 z z z 2 z
> ---- + -- - ---- + ---- + ---- - -- - --- + ----- + ---- + -- + -- - ---- -
11 9 7 5 3 a 12 10 8 6 4 2
a a a a a a a a a a a
7 7 7 7 7 8 8 8 8 9
3 z 4 z 7 z 3 z 3 z 4 z 4 z 3 z 3 z 3 z
> ---- + ---- + ---- - ---- - ---- - ---- - ---- - ---- - ---- - ---- -
11 9 7 5 3 10 8 6 4 9
a a a a a a a a a a
9 9 10 10
5 z 2 z z z
> ---- - ---- - --- - ---
7 5 8 6
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 + 2 q + ----- + - + -- + 5 q t + 4 q t + 8 q t + 6 q t + 8 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 + 8 q t + 6 q t + 7 q t + 4 q t +
16 6 16 7 18 7 18 8 20 8 22 9
> 6 q t + 2 q t + 4 q t + q t + 2 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a222 |
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