0 1
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Polynomial invariants
| Alexander polynomial | 1 |
| Conway polynomial | 1 |
| 2nd Alexander ideal (db, data sources) | [math]\displaystyle{ \{1\} }[/math] |
| Determinant and Signature | { 1, 0 } |
| Jones polynomial | 1 |
| HOMFLY-PT polynomial (db, data sources) | 1 |
| Kauffman polynomial (db, data sources) | 1 |
| The A2 invariant | Data:0 1/QuantumInvariant/A2/1,0 |
| The G2 invariant | [math]\displaystyle{ q^{10}+q^8+q^2+1+ q^{-2} + q^{-8} + q^{-10} }[/math] |
Further Quantum Invariants
Further quantum knot invariants for 0_1.
A1 Invariants.
| Weight | Invariant |
|---|---|
| 1 | [math]\displaystyle{ q+ q^{-1} }[/math] |
| 2 | [math]\displaystyle{ q^2+1+ q^{-2} }[/math] |
| 3 | [math]\displaystyle{ q^3+q+ q^{-1} + q^{-3} }[/math] |
| 4 | [math]\displaystyle{ q^4+q^2+1+ q^{-2} + q^{-4} }[/math] |
| 5 | [math]\displaystyle{ q^5+q^3+q+ q^{-1} + q^{-3} + q^{-5} }[/math] |
| 1 | [math]\displaystyle{ q+ q^{-1} }[/math] |
| 2 | [math]\displaystyle{ q^2+1+ q^{-2} }[/math] |
| 3 | [math]\displaystyle{ q^3+q+ q^{-1} + q^{-3} }[/math] |
A2 Invariants.
| Weight | Invariant |
|---|---|
| 1,1 | [math]\displaystyle{ q^4+2 q^2+2+2 q^{-2} + q^{-4} }[/math] |
| 2,0 | [math]\displaystyle{ q^4+q^2+2+ q^{-2} + q^{-4} }[/math] |
| 3,0 | [math]\displaystyle{ q^6+q^4+2 q^2+2+2 q^{-2} + q^{-4} + q^{-6} }[/math] |
| 1,0 | Data:0 1/QuantumInvariant/A2/1,0 |
| 2,0 | [math]\displaystyle{ q^4+q^2+2+ q^{-2} + q^{-4} }[/math] |
A3 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0 | [math]\displaystyle{ q^4+q^2+2+ q^{-2} + q^{-4} }[/math] |
| 1,0,0 | [math]\displaystyle{ q^3+q+ q^{-1} + q^{-3} }[/math] |
| 1,0,1 | [math]\displaystyle{ q^6+2 q^4+3 q^2+3+3 q^{-2} +2 q^{-4} + q^{-6} }[/math] |
A4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^6+q^4+2 q^2+2+2 q^{-2} + q^{-4} + q^{-6} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^4+q^2+1+ q^{-2} + q^{-4} }[/math] |
B2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ q^4+q^2+ q^{-2} + q^{-4} }[/math] |
| 1,0 | [math]\displaystyle{ q^6+q^2+1+ q^{-2} + q^{-6} }[/math] |
B3 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0 | [math]\displaystyle{ q^{10}+q^6+q^2+1+ q^{-2} + q^{-6} + q^{-10} }[/math] |
B4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^{14}+q^{10}+q^6+q^2+1+ q^{-2} + q^{-6} + q^{-10} + q^{-14} }[/math] |
B5 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0,0 | [math]\displaystyle{ q^{18}+q^{14}+q^{10}+q^6+q^2+1+ q^{-2} + q^{-6} + q^{-10} + q^{-14} + q^{-18} }[/math] |
C3 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0 | [math]\displaystyle{ q^6+q^4+q^2+ q^{-2} + q^{-4} + q^{-6} }[/math] |
C4 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0 | [math]\displaystyle{ q^8+q^6+q^4+q^2+ q^{-2} + q^{-4} + q^{-6} + q^{-8} }[/math] |
C5 Invariants.
| Weight | Invariant |
|---|---|
| 1,0,0,0,0 | [math]\displaystyle{ q^{10}+q^8+q^6+q^4+q^2+ q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10} }[/math] |
D4 Invariants.
| Weight | Invariant |
|---|---|
| 0,1,0,0 | [math]\displaystyle{ q^{10}+q^8+3 q^6+3 q^4+4 q^2+4+4 q^{-2} +3 q^{-4} +3 q^{-6} + q^{-8} + q^{-10} }[/math] |
| 1,0,0,0 | [math]\displaystyle{ q^6+q^4+q^2+2+ q^{-2} + q^{-4} + q^{-6} }[/math] |
G2 Invariants.
| Weight | Invariant |
|---|---|
| 0,1 | [math]\displaystyle{ q^{18}+q^{12}+q^{10}+q^8+q^6+q^2+2+ q^{-2} + q^{-6} + q^{-8} + q^{-10} + q^{-12} + q^{-18} }[/math] |
| 1,0 | [math]\displaystyle{ q^{10}+q^8+q^2+1+ q^{-2} + q^{-8} + q^{-10} }[/math] |
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Computer Talk
The above data is available with the Mathematica package
KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.
(The path below may be different on your system, and possibly also the KnotTheory` date)
In[1]:=
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AppendTo[$Path, "C:/drorbn/projects/KAtlas/"];
<< KnotTheory`
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Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.
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In[3]:=
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K = Knot["0 1"];
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In[4]:=
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Alexander[K][t]
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KnotTheory::loading: Loading precomputed data in PD4Knots`.
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Out[4]=
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1 |
In[5]:=
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Conway[K][z]
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Out[5]=
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1 |
In[6]:=
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Alexander[K, 2][t]
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KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
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Out[6]=
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[math]\displaystyle{ \{1\} }[/math] |
In[7]:=
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{KnotDet[K], KnotSignature[K]}
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Out[7]=
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{ 1, 0 } |
In[8]:=
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Jones[K][q]
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KnotTheory::loading: Loading precomputed data in Jones4Knots`.
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Out[8]=
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1 |
In[9]:=
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HOMFLYPT[K][a, z]
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KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
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Out[9]=
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1 |
In[10]:=
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Kauffman[K][a, z]
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KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
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Out[10]=
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1 |
