 L11n240 |
 L11n242 |
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The 2-Component Link L11n241Visit L11n241's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X8,18,9,17 |
| Gauss Code: | {{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, 6, -9, 8, -2, 11, -6, 7, -8, 9, -7}} |
| Jones Polynomial: | - q-5/2 + 2q-3/2 - 5q-1/2 + 7q1/2 - 9q3/2 + 9q5/2 - 9q7/2 + 7q9/2 - 5q11/2 + 2q13/2 |
| A2 (sl(3)) Invariant: | q-8 + q-6 + q-4 + 3q-2 + q2 - q4 - 2q6 + 2q8 - q10 + 4q12 + q14 + q16 + q18 - 2q20 - q24 |
| HOMFLY-PT Polynomial: | a-7z-1 + a-7z - 4a-5z-1 - 6a-5z - 4a-5z3 - a-5z5 + 6a-3z-1 + 12a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - 5a-1z-1 - 10a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3 |
| Kauffman Polynomial: | a-8 - 3a-8z2 - a-7z-1 + 4a-7z - 5a-7z3 - a-7z5 + 3a-6 - 9a-6z2 + 9a-6z4 - 5a-6z6 - 4a-5z-1 + 18a-5z - 31a-5z3 + 25a-5z5 - 8a-5z7 + 3a-4 - 8a-4z2 + 2a-4z4 + 10a-4z6 - 5a-4z8 - 6a-3z-1 + 31a-3z - 58a-3z3 + 47a-3z5 - 10a-3z7 - a-3z9 + a-2 - 16a-2z4 + 23a-2z6 - 7a-2z8 - 5a-1z-1 + 24a-1z - 41a-1z3 + 26a-1z5 - 3a-1z7 - a-1z9 + 1 + 2z2 - 9z4 + 8z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7 |
| 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, NonAlternating, 241]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 241]] |
Out[4]= | PD[X[12, 1, 13, 2], X[16, 8, 17, 7], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[18, 14, 19, 13], X[22, 20, 11, 19], X[20, 15, 21, 16], > X[14, 21, 15, 22], X[2, 11, 3, 12], X[8, 18, 9, 17]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, -4, 5, -3, 4, 2, -11, -5, 3},
> {10, -1, 6, -9, 8, -2, 11, -6, 7, -8, 9, -7}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(5/2) 2 5 3/2 5/2 7/2 9/2
-q + ---- - ------- + 7 Sqrt[q] - 9 q + 9 q - 9 q + 7 q -
3/2 Sqrt[q]
q
11/2 13/2
> 5 q + 2 q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -8 -6 -4 3 2 4 6 8 10 12 14 16 18
q + q + q + -- + q - q - 2 q + 2 q - q + 4 q + q + q + q -
2
q
20 24
> 2 q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 241]][a, z] |
Out[8]= | 3
1 4 6 5 2 a z 6 z 12 z 10 z 4 z
---- - ---- + ---- - --- + --- + -- - --- + ---- - ---- + 3 a z - ---- +
7 5 3 a z z 7 5 3 a 5
a z a z a z a a a a
3 3 5 5 5 7
10 z 8 z 3 z 5 z 2 z z
> ----- - ---- + a z - -- + ---- - ---- + --
3 a 5 3 a 3
a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 241]][a, z] |
Out[9]= | -8 3 3 -2 1 4 6 5 2 a 4 z 18 z 31 z
1 + a + -- + -- + a - ---- - ---- - ---- - --- - --- + --- + ---- + ---- +
6 4 7 5 3 a z z 7 5 3
a a a z a z a z a a a
2 2 2 3 3 3 3
24 z 2 3 z 9 z 8 z 5 z 31 z 58 z 41 z
> ---- + 7 a z + 2 z - ---- - ---- - ---- - ---- - ----- - ----- - ----- -
a 8 6 4 7 5 3 a
a a a a a a
4 4 4 5 5 5 5
3 4 9 z 2 z 16 z z 25 z 47 z 26 z 5
> 9 a z - 9 z + ---- + ---- - ----- - -- + ----- + ----- + ----- + 5 a z +
6 4 2 7 5 3 a
a a a a a a
6 6 6 7 7 7 8
6 5 z 10 z 23 z 8 z 10 z 3 z 7 8 5 z
> 8 z - ---- + ----- + ----- - ---- - ----- - ---- - a z - 2 z - ---- -
6 4 2 5 3 a 4
a a a a a a
8 9 9
7 z z z
> ---- - -- - --
2 3 a
a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2
2 4 1 1 1 -2 4 3 4 q 4
6 q + 5 q + ----- + ----- + ----- + t + ----- + - + ---- + 5 q t +
6 4 4 3 2 3 2 2 t t
q t q t q t q t
6 6 2 8 2 8 3 10 3 10 4 12 4
> 4 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 3 q t +
14 5
> 2 q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n241 |
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