 L11n107 |
 L11n109 |
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The 2-Component Link L11n108Visit L11n108's page at Knotilus! |
 DrawMorseLink |
| PD Presentation: | X6172 X12,3,13,4 X13,19,14,18 X17,11,18,10 X21,9,22,8 X7,17,8,16 X9,21,10,20 X15,5,16,22 X19,15,20,14 X2536 X4,11,1,12 |
| Gauss Code: | {{1, -10, 2, -11}, {10, -1, -6, 5, -7, 4, 11, -2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8}} |
| Jones Polynomial: | - q-5/2 + q-3/2 - 2q-1/2 - q1/2 + q3/2 - 2q5/2 + 3q7/2 - 3q9/2 + 3q11/2 - 2q13/2 + q15/2 |
| A2 (sl(3)) Invariant: | q-8 + q-6 + 2q-4 + 2q-2 + 4 + 3q2 + q4 + 2q6 - q8 - q10 - 2q12 - 2q14 - q18 + q20 - q24 |
| HOMFLY-PT Polynomial: | a-7z + a-5z-1 - a-5z3 - a-3z-1 - a-3z - a-3z3 - 2a-1z-1 - 5a-1z - 5a-1z3 - a-1z5 + 2az-1 + 3az + az3 |
| Kauffman Polynomial: | a-8 - 4a-8z2 + 4a-8z4 - a-8z6 + 2a-7z - 7a-7z3 + 8a-7z5 - 2a-7z7 - 3a-6z2 + 4a-6z4 + 2a-6z6 - a-6z8 + a-5z-1 + 2a-5z - 12a-5z3 + 13a-5z5 - 3a-5z7 - 3a-4 + 15a-4z2 - 21a-4z4 + 12a-4z6 - 2a-4z8 + a-3z-1 - 3a-3z + 3a-3z3 - 8a-3z5 + 6a-3z7 - a-3z9 + 10a-2z2 - 24a-2z4 + 14a-2z6 - 2a-2z8 - 2a-1z-1 + 4a-1z - 2a-1z3 - 7a-1z5 + 6a-1z7 - a-1z9 + 3 - 4z2 - 3z4 + 5z6 - z8 - 2az-1 + 7az - 10az3 + 6az5 - az7 |
| 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, NonAlternating, 108]]] |
| |
Out[3]= | -Graphics- |
In[4]:= | PD[L = Link[11, NonAlternating, 108]] |
Out[4]= | PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[13, 19, 14, 18], X[17, 11, 18, 10], > X[21, 9, 22, 8], X[7, 17, 8, 16], X[9, 21, 10, 20], X[15, 5, 16, 22], > X[19, 15, 20, 14], X[2, 5, 3, 6], X[4, 11, 1, 12]] |
In[5]:= | GaussCode[L] |
Out[5]= | GaussCode[{1, -10, 2, -11}, {10, -1, -6, 5, -7, 4, 11, -2, -3, 9, -8, 6, -4, 3,
> -9, 7, -5, 8}] |
In[6]:= | Jones[L][q] |
Out[6]= | -(5/2) -(3/2) 2 3/2 5/2 7/2 9/2
-q + q - ------- - Sqrt[q] + q - 2 q + 3 q - 3 q +
Sqrt[q]
11/2 13/2 15/2
> 3 q - 2 q + q |
In[7]:= | A2Invariant[L][q] |
Out[7]= | -8 -6 2 2 2 4 6 8 10 12 14 18
4 + q + q + -- + -- + 3 q + q + 2 q - q - q - 2 q - 2 q - q +
4 2
q q
20 24
> q - q |
In[8]:= | HOMFLYPT[Link[11, NonAlternating, 108]][a, z] |
Out[8]= | 3 3 3 5 1 1 2 2 a z z 5 z z z 5 z 3 z ---- - ---- - --- + --- + -- - -- - --- + 3 a z - -- - -- - ---- + a z - -- 5 3 a z z 7 3 a 5 3 a a a z a z a a a a |
In[9]:= | Kauffman[Link[11, NonAlternating, 108]][a, z] |
Out[9]= | -8 3 1 1 2 2 a 2 z 2 z 3 z 4 z 2
3 + a - -- + ---- + ---- - --- - --- + --- + --- - --- + --- + 7 a z - 4 z -
4 5 3 a z z 7 5 3 a
a a z a z a a a
2 2 2 2 3 3 3 3
4 z 3 z 15 z 10 z 7 z 12 z 3 z 2 z 3 4
> ---- - ---- + ----- + ----- - ---- - ----- + ---- - ---- - 10 a z - 3 z +
8 6 4 2 7 5 3 a
a a a a a a a
4 4 4 4 5 5 5 5
4 z 4 z 21 z 24 z 8 z 13 z 8 z 7 z 5 6
> ---- + ---- - ----- - ----- + ---- + ----- - ---- - ---- + 6 a z + 5 z -
8 6 4 2 7 5 3 a
a a a a a a a
6 6 6 6 7 7 7 7 8
z 2 z 12 z 14 z 2 z 3 z 6 z 6 z 7 8 z
> -- + ---- + ----- + ----- - ---- - ---- + ---- + ---- - a z - z - -- -
8 6 4 2 7 5 3 a 6
a a a a a a a a
8 8 9 9
2 z 2 z z z
> ---- - ---- - -- - --
4 2 3 a
a a a |
In[10]:= | Kh[L][q, t] |
Out[10]= | 2
2 4 1 1 1 -2 1 1 q 2
3 + 3 q + q + ----- + ----- + ----- + t + ----- + - + -- + 2 q t +
6 4 4 4 4 3 2 2 t t
q t q t q t q t
4 6 4 2 6 2 6 3 8 3 10 3
> 2 q t + 2 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t +
8 4 10 4 10 5 12 5 12 6 14 6 16 7
> 2 q t + 2 q t + q t + 2 q t + q t + q t + q t |
| Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n108 |
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