0 1

Polynomial invariants

Alexander polynomial	1
Conway polynomial	1
2nd Alexander ideal (db, data sources)	$\{1\}$
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	$q^{10}+q^{8}+q^{2}+1+q^{-2}+q^{-8}+q^{-10}$

Further Quantum Invariants

Further quantum knot invariants for 0_1.

A1 Invariants.

Weight	Invariant
1	$q+q^{-1}$
2	$q^{2}+1+q^{-2}$
3	$q^{3}+q+q^{-1}+q^{-3}$
4	$q^{4}+q^{2}+1+q^{-2}+q^{-4}$
5	$q^{5}+q^{3}+q+q^{-1}+q^{-3}+q^{-5}$
1	$q+q^{-1}$
2	$q^{2}+1+q^{-2}$
3	$q^{3}+q+q^{-1}+q^{-3}$

A2 Invariants.

Weight	Invariant
1,1	$q^{4}+2q^{2}+2+2q^{-2}+q^{-4}$
2,0	$q^{4}+q^{2}+2+q^{-2}+q^{-4}$
3,0	$q^{6}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}+q^{-6}$
1,0	Data:0 1/QuantumInvariant/A2/1,0
2,0	$q^{4}+q^{2}+2+q^{-2}+q^{-4}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{4}+q^{2}+2+q^{-2}+q^{-4}$
1,0,0	$q^{3}+q+q^{-1}+q^{-3}$
1,0,1	$q^{6}+2q^{4}+3q^{2}+3+3q^{-2}+2q^{-4}+q^{-6}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{6}+q^{4}+2q^{2}+2+2q^{-2}+q^{-4}+q^{-6}$
1,0,0,0	$q^{4}+q^{2}+1+q^{-2}+q^{-4}$

B2 Invariants.

Weight	Invariant
0,1	$q^{4}+q^{2}+q^{-2}+q^{-4}$
1,0	$q^{6}+q^{2}+1+q^{-2}+q^{-6}$

B3 Invariants.

Weight	Invariant
1,0,0	$q^{10}+q^{6}+q^{2}+1+q^{-2}+q^{-6}+q^{-10}$

B4 Invariants.

Weight	Invariant
1,0,0,0	$q^{14}+q^{10}+q^{6}+q^{2}+1+q^{-2}+q^{-6}+q^{-10}+q^{-14}$

B5 Invariants.

Weight	Invariant
1,0,0,0,0	$q^{18}+q^{14}+q^{10}+q^{6}+q^{2}+1+q^{-2}+q^{-6}+q^{-10}+q^{-14}+q^{-18}$

C3 Invariants.

Weight	Invariant
1,0,0	$q^{6}+q^{4}+q^{2}+q^{-2}+q^{-4}+q^{-6}$

C4 Invariants.

Weight	Invariant
1,0,0,0	$q^{8}+q^{6}+q^{4}+q^{2}+q^{-2}+q^{-4}+q^{-6}+q^{-8}$

C5 Invariants.

Weight	Invariant
1,0,0,0,0	$q^{10}+q^{8}+q^{6}+q^{4}+q^{2}+q^{-2}+q^{-4}+q^{-6}+q^{-8}+q^{-10}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{10}+q^{8}+3q^{6}+3q^{4}+4q^{2}+4+4q^{-2}+3q^{-4}+3q^{-6}+q^{-8}+q^{-10}$
1,0,0,0	$q^{6}+q^{4}+q^{2}+2+q^{-2}+q^{-4}+q^{-6}$

G2 Invariants.

Weight	Invariant
0,1	$q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}+2+q^{-2}+q^{-6}+q^{-8}+q^{-10}+q^{-12}+q^{-18}$
1,0	$q^{10}+q^{8}+q^{2}+1+q^{-2}+q^{-8}+q^{-10}$

.

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]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["0 1"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

1

In[5]:=

Conway[K][z]

Out[5]=

1

In[6]:=

Alexander[K, 2][t]

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.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 1, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

1

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

1

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

1