 L11a359 |
 L11a361 |
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 DrawMorseLink |
| PD Presentation: | X12,1,13,2 X14,4,15,3 X22,14,11,13 X16,6,17,5 X18,8,19,7 X20,10,21,9 X2,11,3,12 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,1,21 |
| Gauss Code: | {{1, -7, 2, -8, 4, -9, 5, -10, 6, -11}, {7, -1, 3, -2, 8, -4, 9, -5, 10, -6, 11, -3}} |
| Jones Polynomial: | - q3/2 + q5/2 - 2q7/2 + 2q9/2 - 3q11/2 + 3q13/2 - 3q15/2 + 3q17/2 - 3q19/2 + 2q21/2 - 2q23/2 + q25/2 |
| A2 (sl(3)) Invariant: | q6 + q8 + q10 + q12 + q16 + q20 + q24 + q26 + q28 + q30 - q36 |
| HOMFLY-PT Polynomial: | 4a-9z + 10a-9z3 + 6a-9z5 + a-9z7 - a-7z-1 - 11a-7z - 25a-7z3 - 22a-7z5 - 8a-7z7 - a-7z9 + a-5z-1 + 10a-5z + 15a-5z3 + 7a-5z5 + a-5z7 |
| Kauffman Polynomial: | - a-16z2 + a-15z - 2a-15z3 + a-14z2 - 2a-14z4 + a-13z + 2a-13z3 - 2a-13z5 + 4a-12z4 - 2a-12z6 - a-11z - 2a-11z3 + 6a-11z5 - 2a-11z7 - a-10z2 - 6a-10z4 + 8a-10z6 - 2a-10z8 - 3a-9z + 11a-9z3 - 18a-9z5 + 11a-9z7 - 2a-9z9 + 3a-8z2 - 3a-8z4 - 5a-8z6 + 5a-8z8 - a-8z10 + a-7z-1 - 13a-7z + 42a-7z3 - 48a-7z5 + 21a-7z7 - 3a-7z9 - a-6 + 2a-6z2 + 9a-6z4 - 15a-6z6 + 7a-6z8 - a-6z10 + a-5z-1 - 11a-5z + 25a-5z3 - 22a-5z5 + 8a-5z7 - a-5z9 |
| 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, 360]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 360]] |
Out[4]= | PD[X[12, 1, 13, 2], X[14, 4, 15, 3], X[22, 14, 11, 13], X[16, 6, 17, 5], > X[18, 8, 19, 7], X[20, 10, 21, 9], X[2, 11, 3, 12], X[4, 16, 5, 15], > X[6, 18, 7, 17], X[8, 20, 9, 19], X[10, 22, 1, 21]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -7, 2, -8, 4, -9, 5, -10, 6, -11},
> {7, -1, 3, -2, 8, -4, 9, -5, 10, -6, 11, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2
-q + q - 2 q + 2 q - 3 q + 3 q - 3 q + 3 q -
19/2 21/2 23/2 25/2
> 3 q + 2 q - 2 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | 6 8 10 12 16 20 24 26 28 30 36 q + q + q + q + q + q + q + q + q + q - q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 360]][a, z] |
Out[8]= | 3 3 3 5 5
1 1 4 z 11 z 10 z 10 z 25 z 15 z 6 z 22 z
-(----) + ---- + --- - ---- + ---- + ----- - ----- + ----- + ---- - ----- +
7 5 9 7 5 9 7 5 9 7
a z a z a a a a a a a a
5 7 7 7 9
7 z z 8 z z z
> ---- + -- - ---- + -- - --
5 9 7 5 7
a a a a a |
In[9]:= | Kauffman[Link[11, Alternating, 360]][a, z] |
Out[9]= | 2 2 2
-6 1 1 z z z 3 z 13 z 11 z z z z
-a + ---- + ---- + --- + --- - --- - --- - ---- - ---- - --- + --- - --- +
7 5 15 13 11 9 7 5 16 14 10
a z a z a a a a a a a a a
2 2 3 3 3 3 3 3 4 4
3 z 2 z 2 z 2 z 2 z 11 z 42 z 25 z 2 z 4 z
> ---- + ---- - ---- + ---- - ---- + ----- + ----- + ----- - ---- + ---- -
8 6 15 13 11 9 7 5 14 12
a a a a a a a a a a
4 4 4 5 5 5 5 5 6 6
6 z 3 z 9 z 2 z 6 z 18 z 48 z 22 z 2 z 8 z
> ---- - ---- + ---- - ---- + ---- - ----- - ----- - ----- - ---- + ---- -
10 8 6 13 11 9 7 5 12 10
a a a a a a a a a a
6 6 7 7 7 7 8 8 8 9
5 z 15 z 2 z 11 z 21 z 8 z 2 z 5 z 7 z 2 z
> ---- - ----- - ---- + ----- + ----- + ---- - ---- + ---- + ---- - ---- -
8 6 11 9 7 5 10 8 6 9
a a a a a a a a a a
9 9 10 10
3 z z z z
> ---- - -- - --- - ---
7 5 8 6
a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2 6
6 8 q q 8 10 10 2 12 2 12 3 14 3
2 q + q + -- + -- + q t + q t + 2 q t + q t + q t + 2 q t +
2 t
t
14 4 16 4 16 5 18 5 18 6 20 6 20 7
> 2 q t + q t + q t + 2 q t + 2 q t + 2 q t + q t +
22 7 22 8 24 8 26 9
> q t + q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a360 |
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