0 1

0_1

3_1

Visit 0 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit [${\textrm {KnotilusURL}}({\textrm {GaussCode}}())$ 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	${\textrm {Loop}}(1)$
Gauss code
Dowker-Thistlethwaite code
Conway Notation	Data:0 1/Conway Notation

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 ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(0,1))$ 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)	$\{1\}$
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

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 $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$ $?$

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

\   r    \    j   \	0	χ
1	1	1
-1	1	1

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[0, 1]]
Out[2]=	0
In[3]:=	PD[Knot[0, 1]]
Out[3]=	PD[Loop[1]]
In[4]:=	GaussCode[Knot[0, 1]]
Out[4]=	GaussCode[]
In[5]:=	BR[Knot[0, 1]]
Out[5]=	BR[1, {}]
In[6]:=	alex = Alexander[Knot[0, 1]][t]
Out[6]=	1
In[7]:=	Conway[Knot[0, 1]][z]
Out[7]=	1
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[0, 1], Knot[11, NonAlternating, 34], Knot[11, NonAlternating, 42]}
In[9]:=	{KnotDet[Knot[0, 1]], KnotSignature[Knot[0, 1]]}
Out[9]=	{1, 0}
In[10]:=	J=Jones[Knot[0, 1]][q]
Out[10]=	1
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[0, 1]}
In[12]:=	A2Invariant[Knot[0, 1]][q]
Out[12]=	-2 2 1 + q + q
In[13]:=	Kauffman[Knot[0, 1]][a, z]
Out[13]=	1
In[14]:=	{Vassiliev[2][Knot[0, 1]], Vassiliev[3][Knot[0, 1]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[0, 1]][q, t]
Out[15]=	1 - + q q