0 1

0_1

3_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 0 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 0 1 at Knotilus!
	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

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Data:0 1/BraidPlot Length is Data:0 1/MinimalBraidLength, width is Data:0 1/MinimalBraidWidth, Braid index is Data:0 1/BraidIndex		[{1, 2}, {2, 1}]

[edit Notes on presentations of 0 1]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["0 1"];

In[4]:=

PD[K]

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

Out[4]=

${\textrm {Loop}}(1)$

In[5]:=

GaussCode[K]

Out[5]=

In[6]:=

DTCode[K]

Out[6]=

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

Data:0 1/Conway Notation

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(1,\{\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ Data:0 1/MinimalBraidWidth, Data:0 1/MinimalBraidLength, Data:0 1/BraidIndex }

In[11]:=

Show[BraidPlot[br]]

Data:0 1/BraidPlot

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{1, 2}, {2, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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

"Similar" Knots (within the Atlas)

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

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["0 1"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ 1, 1 }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n34, K11n42,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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.

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 $n+1$-dimensional representation of $sl(2)$)

$n$

$J_{n}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

0_1 3_1