 L11a310 |
 L11a312 |
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The 2-Component Link L11a311Visit L11a311's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,18,21,17 X18,5,19,6 X4,19,5,20 X14,7,15,8 X16,13,17,14 X8,15,1,16 X6,22,7,21 |
| Gauss Code: | {{1, -4, 2, -7, 6, -11, 8, -10}, {4, -1, 3, -2, 9, -8, 10, -9, 5, -6, 7, -5, 11, -3}} |
| Jones Polynomial: | q-15/2 - 4q-13/2 + 8q-11/2 - 14q-9/2 + 19q-7/2 - 23q-5/2 + 22q-3/2 - 20q-1/2 + 15q1/2 - 9q3/2 + 4q5/2 - q7/2 |
| A2 (sl(3)) Invariant: | - q-24 + 3q-20 - q-18 + q-16 + 3q-14 - 4q-12 + 4q-10 + q-8 + 2q-6 + 4q-4 - 4q-2 + 4 - 4q2 - q4 + 3q6 - 2q8 + q10 |
| HOMFLY-PT Polynomial: | - 2a-1z - 2a-1z3 - a-1z5 + 5az + 8az3 + 4az5 + az7 - a3z-1 - 8a3z - 8a3z3 - 3a3z5 + a5z-1 + 4a5z + 3a5z3 - a7z |
| Kauffman Polynomial: | a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 + 3a-1z - 8a-1z3 + 12a-1z5 - 8a-1z7 - 2z4 + 10z6 - 9z8 + 8az - 29az3 + 35az5 - 10az7 - 5az9 + 5a2z2 - 26a2z4 + 42a2z6 - 20a2z8 - a2z10 - a3z-1 + 13a3z - 41a3z3 + 42a3z5 - 3a3z7 - 9a3z9 + a4 + 4a4z2 - 27a4z4 + 41a4z6 - 17a4z8 - a4z10 - a5z-1 + 10a5z - 29a5z3 + 30a5z5 - 5a5z7 - 4a5z9 - 6a6z4 + 12a6z6 - 6a6z8 + 2a7z - 8a7z3 + 10a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6 |
| Khovanov Homology: |
|
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, 311]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, Alternating, 311]] |
Out[4]= | PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[20, 18, 21, 17], X[18, 5, 19, 6], X[4, 19, 5, 20], X[14, 7, 15, 8], > X[16, 13, 17, 14], X[8, 15, 1, 16], X[6, 22, 7, 21]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -4, 2, -7, 6, -11, 8, -10},
> {4, -1, 3, -2, 9, -8, 10, -9, 5, -6, 7, -5, 11, -3}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(15/2) 4 8 14 19 23 22 20
q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] -
13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q]
q q q q q q
3/2 5/2 7/2
> 9 q + 4 q - q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -24 3 -18 -16 3 4 4 -8 2 4 4 2
4 - q + --- - q + q + --- - --- + --- + q + -- + -- - -- - 4 q -
20 14 12 10 6 4 2
q q q q q q q
4 6 8 10
> q + 3 q - 2 q + q |
In[8]:= | HOMFLYPT[Link[11, Alternating, 311]][a, z] |
Out[8]= | 3 5 3
a a 2 z 3 5 7 2 z 3 3 3
-(--) + -- - --- + 5 a z - 8 a z + 4 a z - a z - ---- + 8 a z - 8 a z +
z z a a
5
5 3 z 5 3 5 7
> 3 a z - -- + 4 a z - 3 a z + a z
a |
In[9]:= | Kauffman[Link[11, Alternating, 311]][a, z] |
Out[9]= | 3 5 2
4 a a 3 z 3 5 7 2 z 2 2
a - -- - -- + --- + 8 a z + 13 a z + 10 a z + 2 a z - ---- + 5 a z +
z z a 2
a
3 3
4 2 8 2 z 8 z 3 3 3 5 3 7 3
> 4 a z - a z + -- - ---- - 29 a z - 41 a z - 29 a z - 8 a z -
3 a
a
4 5 5
4 5 z 2 4 4 4 6 4 8 4 z 12 z
> 2 z + ---- - 26 a z - 27 a z - 6 a z + 2 a z - -- + ----- +
2 3 a
a a
6
5 3 5 5 5 7 5 6 4 z 2 6
> 35 a z + 42 a z + 30 a z + 10 a z + 10 z - ---- + 42 a z +
2
a
7
4 6 6 6 8 6 8 z 7 3 7 5 7
> 41 a z + 12 a z - a z - ---- - 10 a z - 3 a z - 5 a z -
a
7 7 8 2 8 4 8 6 8 9 3 9
> 4 a z - 9 z - 20 a z - 17 a z - 6 a z - 5 a z - 9 a z -
5 9 2 10 4 10
> 4 a z - a z - a z |
In[10]:= | Kh[L][q, t] |
Out[10]= | 10 1 3 1 5 3 9 6 11
11 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- +
2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3
q q t q t q t q t q t q t q t q t
8 12 11 10 12 2 2 2 4 2
> ----- + ----- + ----- + ---- + ---- + 6 t + 9 q t + 3 q t + 6 q t +
6 3 6 2 4 2 4 2
q t q t q t q t q t
4 3 6 3 8 4
> q t + 3 q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a311 |
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