 L11n32 |
 L11n34 |
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 DrawMorseLink |
| PD Presentation: | X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19 |
| Gauss Code: | {{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}} |
| Jones Polynomial: | q-9/2 - 2q-7/2 + q-5/2 - q-3/2 - q-1/2 + q1/2 - 2q3/2 + 3q5/2 - 3q7/2 + 2q9/2 - q11/2 |
| A2 (sl(3)) Invariant: | - q-14 + 2q-8 + 2q-6 + 2q-4 + 3q-2 + q2 - q4 - q6 - q10 + q12 + q16 + q18 |
| HOMFLY-PT Polynomial: | - a-5z-1 - a-5z + 3a-3z-1 + 5a-3z + 2a-3z3 - 4a-1z-1 - 8a-1z - 5a-1z3 - a-1z5 + 2az-1 + 6az + 5az3 + az5 - 2a3z - a3z3 |
| Kauffman Polynomial: | - a-5z-1 + 4a-5z - 7a-5z3 + 5a-5z5 - a-5z7 - a-4 + 7a-4z2 - 13a-4z4 + 10a-4z6 - 2a-4z8 - 3a-3z-1 + 16a-3z - 24a-3z3 + 11a-3z5 + 2a-3z7 - a-3z9 - 3a-2 + 17a-2z2 - 34a-2z4 + 23a-2z6 - 4a-2z8 - 4a-1z-1 + 24a-1z - 40a-1z3 + 17a-1z5 + 2a-1z7 - a-1z9 - 2 + 12z2 - 28z4 + 19z6 - 3z8 - 2az-1 + 17az - 34az3 + 21az5 - 3az7 - a2 - 3a2z4 + 5a2z6 - a2z8 + 5a3z - 11a3z3 + 10a3z5 - 2a3z7 - 2a4z2 + 4a4z4 - a4z6 |
| 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, 33]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 33]] |
Out[4]= | PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[9, 14, 10, 15], > X[8, 4, 9, 3], X[5, 13, 6, 12], X[13, 5, 14, 22], X[15, 18, 16, 19], > X[11, 17, 12, 16], X[17, 11, 18, 10], X[2, 20, 3, 19]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8,
> 11, -2, 3, 7}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(9/2) 2 -(5/2) -(3/2) 1 3/2 5/2
q - ---- + q - q - ------- + Sqrt[q] - 2 q + 3 q -
7/2 Sqrt[q]
q
7/2 9/2 11/2
> 3 q + 2 q - q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -14 2 2 2 3 2 4 6 10 12 16 18
-q + -- + -- + -- + -- + q - q - q - q + q + q + q
8 6 4 2
q q q q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 33]][a, z] |
Out[8]= | 3 3
1 3 4 2 a z 5 z 8 z 3 2 z 5 z
-(----) + ---- - --- + --- - -- + --- - --- + 6 a z - 2 a z + ---- - ---- +
5 3 a z z 5 3 a 3 a
a z a z a a a
5
3 3 3 z 5
> 5 a z - a z - -- + a z
a |
In[9]:= | Kauffman[Link[11, NonAlternating, 33]][a, z] |
Out[9]= | -4 3 2 1 3 4 2 a 4 z 16 z 24 z
-2 - a - -- - a - ---- - ---- - --- - --- + --- + ---- + ---- + 17 a z +
2 5 3 a z z 5 3 a
a a z a z a a
2 2 3 3 3
3 2 7 z 17 z 4 2 7 z 24 z 40 z 3
> 5 a z + 12 z + ---- + ----- - 2 a z - ---- - ----- - ----- - 34 a z -
4 2 5 3 a
a a a a
4 4 5 5
3 3 4 13 z 34 z 2 4 4 4 5 z 11 z
> 11 a z - 28 z - ----- - ----- - 3 a z + 4 a z + ---- + ----- +
4 2 5 3
a a a a
5 6 6 7
17 z 5 3 5 6 10 z 23 z 2 6 4 6 z
> ----- + 21 a z + 10 a z + 19 z + ----- + ----- + 5 a z - a z - -- +
a 4 2 5
a a a
7 7 8 8 9 9
2 z 2 z 7 3 7 8 2 z 4 z 2 8 z z
> ---- + ---- - 3 a z - 2 a z - 3 z - ---- - ---- - a z - -- - --
3 a 4 2 3 a
a a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 3 2 1 1 1 1 1 1 2
4 + -- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- +
2 10 5 8 4 6 4 6 3 4 3 6 2 4 2
q q t q t q t q t q t q t q t
1 2 1 2 2 4 2 2 4 2 6 2
> ----- + - + ---- + ---- + 2 t + 3 q t + q t + 2 q t + 2 q t + q t +
2 2 t 4 2
q t q t q t
4 3 6 3 6 4 8 4 8 5 10 5 12 6
> 2 q t + 2 q t + q t + 2 q t + q t + q t + q t |
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