 L11a14 |
 L11a16 |
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 Knotscape |
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The 2-Component Link L11a15Visit L11a15's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,7,13,8 X4,13,1,14 X18,10,19,9 X8493 X14,6,15,5 X22,16,5,15 X20,18,21,17 X16,22,17,21 X10,20,11,19 X2,12,3,11 |
| Gauss Code: | {{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -8, 9, -7}} |
| Jones Polynomial: | q-5/2 - 3q-3/2 + 5q-1/2 - 9q1/2 + 11q3/2 - 14q5/2 + 13q7/2 - 12q9/2 + 9q11/2 - 6q13/2 + 4q15/2 - q17/2 |
| A2 (sl(3)) Invariant: | - q-8 + q-6 - q-2 + 4 + q4 + 3q6 + q8 + 4q10 + q12 + q14 + q16 - 4q18 - q22 - 2q24 + q26 |
| HOMFLY-PT Polynomial: | a-7z-1 - a-7z3 - 2a-5z-1 - a-5z + a-5z3 + a-5z5 - a-3z + a-3z5 + a-1z-1 + 3a-1z + 2a-1z3 + a-1z5 - az - az3 |
| Kauffman Polynomial: | - a-9z3 + 3a-9z5 - a-9z7 + 2a-8 + a-8z2 - 15a-8z4 + 16a-8z6 - 4a-8z8 - a-7z-1 + 6a-7z3 - 21a-7z5 + 20a-7z7 - 5a-7z9 + 5a-6 - a-6z2 - 24a-6z4 + 21a-6z6 - 2a-6z10 - 2a-5z-1 + a-5z + 13a-5z3 - 36a-5z5 + 33a-5z7 - 9a-5z9 + 3a-4 - 2a-4z2 - 9a-4z4 + 11a-4z6 - 2a-4z10 - 2a-3z + 9a-3z3 - 9a-3z5 + 8a-3z7 - 4a-3z9 - a-2 - a-2z2 + 4a-2z4 + 2a-2z6 - 4a-2z8 + a-1z-1 - 5a-1z + 7a-1z3 - 4a-1z7 + 3z4 - 4z6 - 2az + 4az3 - 3az5 + a2z2 - a2z4 |
| 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, 15]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 15]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[18, 10, 19, 9], > X[8, 4, 9, 3], X[14, 6, 15, 5], X[22, 16, 5, 15], X[20, 18, 21, 17], > X[16, 22, 17, 21], X[10, 20, 11, 19], X[2, 12, 3, 11]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4,
> 10, -8, 9, -7}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(5/2) 3 5 3/2 5/2 7/2 9/2
q - ---- + ------- - 9 Sqrt[q] + 11 q - 14 q + 13 q - 12 q +
3/2 Sqrt[q]
q
11/2 13/2 15/2 17/2
> 9 q - 6 q + 4 q - q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -8 -6 -2 4 6 8 10 12 14 16 18 22
4 - q + q - q + q + 3 q + q + 4 q + q + q + q - 4 q - q -
24 26
> 2 q + q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 15]][a, z] |
Out[8]= | 3 3 3 5 5 5 1 2 1 z z 3 z z z 2 z 3 z z z ---- - ---- + --- - -- - -- + --- - a z - -- + -- + ---- - a z + -- + -- + -- 7 5 a z 5 3 a 7 5 a 5 3 a a z a z a a a a a a |
In[9]:= | Kauffman[Link[11, Alternating, 15]][a, z] |
Out[9]= | 2 2
2 5 3 -2 1 2 1 z 2 z 5 z z z
-- + -- + -- - a - ---- - ---- + --- + -- - --- - --- - 2 a z + -- - -- -
8 6 4 7 5 a z 5 3 a 8 6
a a a a z a z a a a a
2 2 3 3 3 3 3
2 z z 2 2 z 6 z 13 z 9 z 7 z 3 4
> ---- - -- + a z - -- + ---- + ----- + ---- + ---- + 4 a z + 3 z -
4 2 9 7 5 3 a
a a a a a a
4 4 4 4 5 5 5 5
15 z 24 z 9 z 4 z 2 4 3 z 21 z 36 z 9 z
> ----- - ----- - ---- + ---- - a z + ---- - ----- - ----- - ---- -
8 6 4 2 9 7 5 3
a a a a a a a a
6 6 6 6 7 7 7 7
5 6 16 z 21 z 11 z 2 z z 20 z 33 z 8 z
> 3 a z - 4 z + ----- + ----- + ----- + ---- - -- + ----- + ----- + ---- -
8 6 4 2 9 7 5 3
a a a a a a a a
7 8 8 9 9 9 10 10
4 z 4 z 4 z 5 z 9 z 4 z 2 z 2 z
> ---- - ---- - ---- - ---- - ---- - ---- - ----- - -----
a 8 2 7 5 3 6 4
a a a a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 1 2 1 2 3 2 4 4 2
6 + 5 q + ----- + ----- + ----- + - + ---- + 7 q t + 4 q t + 7 q t +
6 3 4 2 2 2 t 2
q t q t q t q t
6 2 6 3 8 3 8 4 10 4 10 5 12 5
> 7 q t + 6 q t + 7 q t + 6 q t + 6 q t + 3 q t + 6 q t +
12 6 14 6 14 7 16 7 18 8
> 3 q t + 3 q t + q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a15 |
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