0 1

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0_1

3_1

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 The Coloured Jones Polynomials
9 Computer Talk
10 Modifying This Page

(KnotPlot image)

See the full Rolfsen Knot Table.

Visit 0 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit $KnotilusURL(GaussCode())$ at Knotilus!

Visit 0 1's page at the original Knot Atlas!

[edit 0 1 Quick Notes]

Also known as "the Unknot"

[edit 0 1 Further Notes and Views]

A temple symbol MANJI in a Japanese map[1]

A toroidal bubble in glass [2]

Simple closed loop as pseudo-knot

Elaborate heraldic depiction

[edit] Knot presentations

Planar diagram presentation	$Loop(1)$
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]= $Loop(1)$

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]= $BR(1,{})$

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-

[edit] 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]

[edit] Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	Missing
Rasmussen s-Invariant	Missing

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

[edit] 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	$q 4 + 2 q 2 + 2 + 2 q -2 + q -4$
2,0	$q 4 + q 2 + 2 + q -2 + q -4$
3,0	$q 6 + q 4 + 2 q 2 + 2 + 2 q -2 + q -4 + q -6$
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	$q 6 + 2 q 4 + 3 q 2 + 3 + 3 q -2 + 2 q -4 + q -6$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q 6 + q 4 + 2 q 2 + 2 + 2 q -2 + q -4 + q -6$
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	$q 10 + q 8 + 3 q 6 + 3 q 4 + 4 q 2 + 4 + 4 q -2 + 3 q -4 + 3 q -6 + q -8 + q -10$
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.

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

[edit] "Similar" Knots (within the Atlas)

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

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

Computer Talk

(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.

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]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(0, 0)

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = 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

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n + 1

-dimensional representation of

s l (2)

)

n

J n

[edit] 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.

[edit] 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

Retrieved from "http://katlas.org/wiki/0_1"

Category: Knot Page

0 1

From Knot Atlas

Contents

[edit] Knot presentations

[edit] Three dimensional invariants

[edit] Four dimensional invariants

[edit] Polynomial invariants

[edit] "Similar" Knots (within the Atlas)

[edit] Vassiliev invariants

[edit] Khovanov Homology

[edit] The Coloured Jones Polynomials

[edit] Computer Talk

[edit] Modifying This Page

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