Unknot

Unknot
Common nameCircle
Arf invariant0
Braid no.1
Bridge no.0
Crossing no.0
Genus0
Linking no.0
Stick no.3
Tunnel no.0
Unknotting no.0
Conway notation-
A–B notation01
Dowker notation-
Next31
Other
torus, fibered, prime, slice, fully amphichiral
Two simple diagrams of the unknot

In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic (that is, deformable) to a geometrically round circle, the standard unknot.

The unknot is the only knot that is the boundary of an embedded disk, which gives the characterization that only unknots have Seifert genus 0. Similarly, the unknot is the identity element with respect to the knot sum operation.

Unknotting problem

Main article: Unknotting problem

Deciding if a particular knot is the unknot was a major driving force behind knot invariants, since it was thought this approach would possibly give an efficient algorithm to recognize the unknot from some presentation such as a knot diagram. Unknot recognition is known to be in both NP and co-NP.

It is known that knot Floer homology and Khovanov homology detect the unknot, but these are not known to be efficiently computable for this purpose. It is not known whether the Jones polynomial or finite type invariants can detect the unknot.

Examples

It can be difficult to find a way to untangle string even though the fact it started out untangled proves the task is possible. Thistlethwaite and Ochiai provided many examples of diagrams of unknots that have no obvious way to simplify them, requiring one to temporarily increase the diagram's crossing number.

While rope is generally not in the form of a closed loop, sometimes there is a canonical way to imagine the ends being joined together. From this point of view, many useful practical knots are actually the unknot, including those that can be tied in a bight.[1]

Every tame knot can be represented as a linkage, which is a collection of rigid line segments connected by universal joints at their endpoints. The stick number is the minimal number of segments needed to represent a knot as a linkage, and a stuck unknot is a particular unknotted linkage that cannot be reconfigured into a flat convex polygon.[2] Like crossing number, a linkage might need to be made more complex by subdividing its segments before it can be simplified.

Invariants

The Alexander–Conway polynomial and Jones polynomial of the unknot are trivial:

No other knot with 10 or fewer crossings has trivial Alexander polynomial, but the Kinoshita–Terasaka knot and Conway knot (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.

The unknot is the only knot whose knot group is an infinite cyclic group, and its knot complement is homeomorphic to a solid torus.

See also

  • Knot (mathematics) – Embedding of the circle in three dimensional Euclidean space
  • Unlink – Link that consists of finitely many unlinked unknots
  • Unknotting number – Minimum number of times a specific knot must be passed through itself to become untied

References

  1. ^ Volker Schatz. "Knotty topics". Archived from the original on 2011-07-17. Retrieved 2007-04-23.
  2. ^ Godfried Toussaint (2001). "A new class of stuck unknots in Pol-6" (PDF). Contributions to Algebra and Geometry. 42 (2): 301–306. Archived from the original (PDF) on 2003-05-12.

Read other articles:

La Salle University

Questa voce sugli argomenti università della Pennsylvania e Filadelfia è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. La Salle University UbicazioneStato Stati Uniti CittàFiladelfia Dati generaliSoprannomeExplorers MottoVirtus Scientia Fondazione1863 TipoUniversità privata RettoreMichael McGinniss PresidenteColleen Hanycz Studenti7 554 Dipendenti373 Colori          Blu e Oro Mappa di localizzazione...

 

Pomadasys

Pomadasys Pomadasys kaakan Klasifikasi ilmiah Domain: Eukaryota Kerajaan: Animalia Filum: Chordata Kelas: Actinopterygii Ordo: Perciformes Famili: Haemulidae Subfamili: Haemulinae Genus: PomadasysLacépède, 1802 Spesies tipe Pomadasys argenteus(Forsskål, 1775) Spesies Lihat teks Sinonim[1] Anomalodon S. Bowdich, 1825 Dacymba D. S. Jordan & C. L. Hubbs, 1917 Polotus Blyth, 1858 Pomadasyina Fowler, 1931 Pristipoma Quoy & Gaimard, 1824 Pristipomus Oken, 1817 Rhencus D. S. Jord...

 

Aqua (satellite)

NASA scientific research satellite AquaAqua (EOS PM-1)Mission typeEarth observationOperatorNASACOSPAR ID2002-022A SATCAT no.27424Websiteaqua.nasa.govMission duration6 years (planned)Elapsed: 21 years, 11 months, 4 days Spacecraft propertiesBusT330 (AB-1200)ManufacturerTRWLaunch mass3,117 kilograms (6,872 lb)Dimensions4.81 m × 16.7 m × 8.04 m (15.8 ft × 54.8 ft × 26.4 ft)Power4.444 kilowatts Start of missionLaunch dateMa...

Sir Ronald Cross, 1st Baronet

British politician and diplomat For other people named Ronald Cross, see Ronald Cross (disambiguation). The Right HonourableSir Ronald CrossBt KCMG KCVO17th Governor of TasmaniaIn office22 August 1951 – 4 June 1958MonarchsGeorge VI Elizabeth IIPremierRobert CosgrovePreceded byHugh BinneySucceeded byThomas Corbett, 2nd Baron RowallanMember of Parliamentfor OrmskirkIn office25 February 1950 – 5 April 1951Preceded byHarold WilsonSucceeded byArthur Salter Persona...

 

Sérigraphie

Procédé de reproduction sérigraphique La sérigraphie (du latin sericum la soie et du grec graphein l’écriture) est une technique d’imprimerie qui utilise des pochoirs (à l'origine, des écrans de soie) interposés entre l’encre et le support. Les supports sur lesquels la sérigraphie est appliquée peuvent être variés (papier, carton, textile, métal, verre, bois, etc.). Le champ d'application de la sérigraphie est très large : signalétique, objets de communication, créati...

 

Konfederasi Sepak Bola Amerika Selatan

South American Football ConfederationSingkatanCONMEBOLCSFTanggal pendirian9 Juli 1916; 107 tahun lalu (1916-07-09)TipeOrganisasi olahragaKantor pusatLuque (Gran Asunción), ParaguayKoordinat25°15′38″S 57°30′58″W / 25.26056°S 57.51611°W / -25.26056; -57.51611Wilayah layanan Amerika SelatanJumlah anggota 10 asosiasiBahasa resmi Spanyol PortugisPresidenAlejandro DomínguezWakil Presiden Laureano González Claudio Tapia Ramón Jesurún[1] Sekretari...

Ramlila

Folk re-enactment of the life of Hindu deity Rama Ramleela redirects here. For the 2017 film, see Ramaleela. For other uses, see Ramleela (disambiguation). This image is taken during 2018 World Famous Ramnagar Ramlila Part of a series onHinduism Hindus History OriginsHistorical Hindu synthesis (500/200 BCE-300 CE) History Indus Valley Civilisation Historical Vedic religion Dravidian folk religion Śramaṇa Tribal religions in India Traditional Itihasa-Purana Epic-Puranic royal genealogies Ep...

 

Voya Financial

American financial services company Voya Financial, Inc.Headquarters at Helmsley BuildingCompany typePublic companyTraded asNYSE: VOYAS&P 400 componentIndustryFinancial servicesPredecessorING U.S.Founded1991; 33 years ago (1991)(As an ING subsidiary)HeadquartersHelmsley BuildingNew York City, U.S.Key peopleRodney O. Martin, Jr.(Chairman & CEO)Michael S. Smith(CFO)ProductsRetirement servicesInsuranceAsset managementEmployee benefitsRevenue US$7.6 billion (2020)Op...

 

Traités de Westphalie

Traités de Westphalie Bartholomeus van der Helst.- Banquet de la garde civile d'Amsterdam fêtant la paix de Münster (1648), exposé au Rijksmuseum Amsterdam. Données clés Type de traité Traité de paix Langue Français Données clés Signé 24 octobre 1648Mairie de Münster, principauté épiscopale de Münster Parties Parties Saint-Empire romain germanique Royaume de France Provinces-Unies  Monarchie espagnole  Royaume de Suède  Danemark-Norvège Confédération suisse...

2012 United States presidential election in North Dakota

Election in North Dakota Main article: 2012 United States presidential election 2012 United States presidential election in North Dakota ← 2008 November 6, 2012 2016 →   Nominee Mitt Romney Barack Obama Party Republican Democratic–NPL Home state Massachusetts Illinois Running mate Paul Ryan Joe Biden Electoral vote 3 0 Popular vote 188,163 124,827 Percentage 58.32% 38.69% County Results Romney   40-50%   50-60%   6...

 

Indiana–Michigan State football rivalry

American college football rivalry The topic of this article may not meet Wikipedia's general notability guideline. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted.Find sources: Indiana–Michigan State football rivalry – news · newspape...

 

Gujranwala

For other uses, see Gujranwala (disambiguation). Metropolis in PunjabGujranwala گوجرانوالہپہلوانوں کا شہرMetropolis From top, left to right: Nishan-e-Manzil, UCP Gujranwala, Jinnah Interchange flyover, Mall of GujranwalaNickname: City of Wrestlers[1]GujranwalaLocation in PakistanCoordinates: 32°9′24″N 74°11′24″E / 32.15667°N 74.19000°E / 32.15667; 74.19000Country PakistanProvince PunjabDivisionGujranwalaDistrict...

History of the Boston Celtics

Sports team history The TD Garden, seen here during the 2008 NBA Finals, has been the Celtics' arena since 1995. The Boston Celtics are an American professional basketball team based in Boston. Founded in 1946 as a charter member of the Basketball Association of America, the Celtics then moved into the National Basketball Association (NBA) in 1949, as said league was formed by the merger of the BAA and the National Basketball League. Currently playing in the Atlantic Division of the Eastern C...

 

Institute of Contemporary Arts

Art and cultural centre in London For other institutions of this name, see Institute of Contemporary Art. Institute of Contemporary ArtsLocation within Central LondonEstablished1946LocationThe Mall, London(offices in Carlton House Terrace)Coordinates51°30′24″N 0°07′50″W / 51.506608°N 0.13061°W / 51.506608; -0.13061DirectorBengi UnsalPublic transit access Charing Cross Charing CrossWebsitewww.ica.art The Institute of Contemporary Arts (ICA) is an artistic an...

 

List of Legislative Council of Hong Kong members 1991–1995

1843–1941 Legco1946–1985 Legco1985–1988 LegCo(1985)1988–1991 LegCo(1988)1991–1995 LegCo(1991)1995–1997 LegCo(1995)Provisional LegCo(1996)1st LegCo(1998)2nd LegCo(2000)3rd LegCo(2004)4th LegCo(2008)5th LegCo(2012)6th LegCo(2016)7th LegCo(2021) This is a list of Members elected to the Legislative Council in the colonial period at the 1991 election, held on 15 September 1991. Composition Affiliation Election At dissolution Co-operative Resources Centre/Liberal Party 0 15 Liberal Dem...

German submarine U-505

German World War II submarine USS Nemo redirects here. For the submersible observation vessel, see DSV-5 Nemo. U-505 shortly after being captured, pictured from the USS Pillsbury in preparation for towing History Nazi Germany NameU-505 Ordered25 September 1939 BuilderDeutsche Werft AG, Hamburg-Finkenwerder Yard number295 Laid down12 June 1940 Launched24 May 1941 Commissioned26 August 1941 FateCaptured by US Navy on 4 June 1944[1][2] StatusPreserved as a museum ship[2] ...

 

Dimetindene

Chemical compound DimetindeneClinical dataTrade namesFenistilAHFS/Drugs.comInternational Drug NamesRoutes ofadministrationOral, topicalATC codeD04AA13 (WHO) R06AB03 (WHO)Legal statusLegal status AU: S4 (Prescription only) UK: General sales list (GSL, OTC) Identifiers IUPAC name N,N-Dimethyl-2-[3-(1-pyridin-2-ylethyl)-1H-inden-2-yl]ethan-1-amine CAS Number5636-83-9 Y 3614-69-5PubChem CID21855DrugBankDB08801 YChemSpider20541 YUNII661FH77Z3PKEGGD07853 ...

 

Montbeugny

Montbeugny La mairie. Administration Pays France Région Auvergne-Rhône-Alpes Département Allier Arrondissement Moulins Intercommunalité Communauté d'agglomération Moulins Communauté Maire Mandat Béatrice Genty 2023-2026 Code postal 03340 Code commune 03180 Démographie Gentilé Montbeunois Populationmunicipale 651 hab. (2021 ) Densité 20 hab./km2 Géographie Coordonnées 46° 31′ 47″ nord, 3° 29′ 23″ est Altitude Min. 238 mMax. 2...

Черепахин, Борис Борисович

В Википедии есть статьи о других людях с фамилией Черепахин. Борис Борисович Черепахин Дата рождения 17 (29) ноября 1894(1894-11-29) Место рождения село Белокриница, Кременецкий уезд, Волынская губерния, Российская империя Дата смерти 27 августа 1969(1969-08-27) (74 года) Место смерти �...

 

Sterling Hill Historic District (Plainfield and Sterling, Connecticut)

Historic district in Connecticut, United States United States historic placeSterling Hill Historic DistrictU.S. National Register of Historic PlacesU.S. Historic district First Baptist Church of SterlingShow map of ConnecticutShow map of the United StatesLocationSterling Hill Rd. and CT 14A, Plainfield and Sterling, ConnecticutCoordinates41°41′23″N 71°50′56″W / 41.68972°N 71.84889°W / 41.68972; -71.84889Area25 acres (10 ha)Architectural styleGreek...