0 1

0_1

3_1

Visit 0 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit [[math]\displaystyle{ \textrm{KnotilusURL}(\textrm{GaussCode}()) }[/math] 0 1's page] at Knotilus! Visit 0 1's page at the original Knot Atlas! Also known as "the Unknot"

A temple symbol MANJI in a Japanese map[1]		A toroidal bubble in glass [2]		Simple closed loop as pseudo-knot
Emblem of Fukuoka prefecture, Japan		Elaborate heraldic depiction		Ornamentation in Palermo, Sicily

Knot presentations

Planar diagram presentation	[math]\displaystyle{ \textrm{Loop}(1) }[/math]
Gauss code
Dowker-Thistlethwaite code
Conway Notation	Data:0 1/Conway Notation

Minimum Braid Representative: Length is 0, width is 1. Braid index is 1.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Unknotting number 0 3-genus 0 Bridge index 1 Super bridge index Nakanishi index Maximal Thurston-Bennequin number [-1][-1] Hyperbolic Volume Data:0 1/HyperbolicVolume A-Polynomial See Data:0 1/A-polynomial

[edit Notes for 0 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(0,1)) }[/math] Rasmussen s-Invariant Missing

[edit Notes for 0 1's four dimensional invariants]

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]

.

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]=

[math]\displaystyle{ \{1\} }[/math]

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n34, K11n42, ...}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {...}

Vassiliev invariants

V2 and V3:

(0, 0)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math] [math]\displaystyle{ ? }[/math]

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of 0 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:0 1/KhovanovTable

Integral Khovanov Homology (db, data source)

Data:0 1/Integral Khovanov Homology

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the [math]\displaystyle{ n+1 }[/math]-dimensional representation of [math]\displaystyle{ sl(2) }[/math])

[math]\displaystyle{ n }[/math]	[math]\displaystyle{ J_n }[/math]
2	Not Available
3	Not Available
4	Not Available
5	Not Available
6	Not Available
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[0, 1]]
Out[2]=	PD[Loop[1]]
In[3]:=	GaussCode[Knot[0, 1]]
Out[3]=	GaussCode[]
In[4]:=	DTCode[Knot[0, 1]]
Out[4]=	DTCode[]
In[5]:=	br = BR[Knot[0, 1]]
Out[5]=	BR[1, {}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{1, 0}
In[7]:=	BraidIndex[Knot[0, 1]]
Out[7]=	1
In[8]:=	Show[DrawMorseLink[Knot[0, 1]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[0, 1]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{, 0, 0, 1, NotAvailable, NotAvailable}
In[10]:=	alex = Alexander[Knot[0, 1]][t]
Out[10]=	1
In[11]:=	Conway[Knot[0, 1]][z]
Out[11]=	1
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[0, 1], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}
In[13]:=	{KnotDet[Knot[0, 1]], KnotSignature[Knot[0, 1]]}
Out[13]=	{1, 0}
In[14]:=	Jones[Knot[0, 1]][q]
Out[14]=	1
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[0, 1]}
In[16]:=	A2Invariant[Knot[0, 1]][q]
Out[16]=	-2 2 1 + q + q
In[17]:=	HOMFLYPT[Knot[0, 1]][a, z]
Out[17]=	1
In[18]:=	Kauffman[Knot[0, 1]][a, z]
Out[18]=	1
In[19]:=	{Vassiliev[2][Knot[0, 1]], Vassiliev[3][Knot[0, 1]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[0, 1]][q, t]
Out[20]=	1 - + q q

See/edit the Rolfsen_Splice_Template.