[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/invertible-knot-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/invertible-knot-wikipedia\/","headline":"Invertible knot – Wikipedia","name":"Invertible knot – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In mathematics, especially in the area of topology known as knot theory, an","datePublished":"2022-08-08","dateModified":"2022-08-08","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1e\/Knot-trefoil-dir-128.png\/90px-Knot-trefoil-dir-128.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1e\/Knot-trefoil-dir-128.png\/90px-Knot-trefoil-dir-128.png","height":"92","width":"90"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/invertible-knot-wikipedia\/","about":["Wiki"],"wordCount":2390,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, especially in the area of topology known as knot theory, an invertible knot is a knot that can be continuously deformed to itself, but with its orientation reversed.  A non-invertible knot is any knot which does not have this property.  The invertibility of a knot is a knot invariant. An invertible link is the link equivalent of an invertible knot.There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.[1]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Table of ContentsBackground[edit]Invertible knots[edit]Strongly invertible knots[edit]Non-invertible knots[edit]See also[edit]References[edit]External links[edit]Background[edit]Number of invertible and non-invertible knots for each crossing numberNumber of crossings345678910111213141516OEIS sequenceNon-invertible knots00000123318711446919381182265811309875A052402Invertible knots1123720471323651032306988542671278830A052403It has long been known that most of the simple knots, such as the trefoil knot and the figure-eight knot are invertible. In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.[2] It is now known almost all knots are non-invertible.[3]Invertible knots[edit]         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4The simplest non-trivial invertible knot, the trefoil knot. Rotating the knot 180 degrees in 3-space about an axis in the plane of the diagram produces the same knot diagram, but with the arrow’s direction reversed.All knots with crossing number of 7 or less are known to be invertible. No general method is known that can distinguish if a given knot is invertible.[4] The problem can be translated into algebraic terms,[5] but unfortunately there is no known algorithm to solve this algebraic problem.If a knot is invertible and amphichiral, it is fully amphichiral. The simplest knot with this property is the figure eight knot. A chiral knot that is invertible is classified as a reversible knot.[6]Strongly invertible knots[edit]A more abstract way to define an invertible knot is to say there is an orientation-preserving homeomorphism of the 3-sphere which takes the knot to itself but reverses the orientation along the knot.  By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere,  we arrive at the definition of a strongly invertible knot.  All knots with tunnel number one, such as the trefoil knot and figure-eight knot, are strongly invertible.[7]Non-invertible knots[edit]  The non-invertible knot 817, the simplest of the non-invertible knots.The simplest example of a non-invertible knot is the knot 817 (Alexander-Briggs notation) or .2.2 (Conway notation). The pretzel knot 7,\u00a05,\u00a03 is non-invertible, as are all pretzel knots of the form (2p\u00a0+\u00a01),\u00a0(2q\u00a0+\u00a01),\u00a0(2r\u00a0+\u00a01), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.[2]See also[edit]References[edit]^ Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998), “The first 1,701,936 knots” (PDF), The Mathematical Intelligencer, 20 (4): 33\u201348, doi:10.1007\/BF03025227, MR\u00a01646740, archived from the original (PDF) on 2013-12-15.^ a b Trotter, H. F. (1963), “Non-invertible knots exist”, Topology, 2: 275\u2013280, doi:10.1016\/0040-9383(63)90011-9, MR\u00a00158395.^ Murasugi, Kunio (2007), Knot Theory and Its Applications, Springer, p.\u00a045, ISBN\u00a09780817647186.^ Weisstein, Eric W. “Invertible Knot”. MathWorld. Accessed: May 5, 2013.^ Kuperberg, Greg (1996), “Detecting knot invertibility”, Journal of Knot Theory and its Ramifications, 5 (2): 173\u2013181, arXiv:q-alg\/9712048, doi:10.1142\/S021821659600014X, MR\u00a01395778.^ Clark, W. Edwin; Elhamdadi, Mohamed; Saito, Masahico; Yeatman, Timothy (2013), Quandle colorings of knots and applications, arXiv:1312.3307, Bibcode:2013arXiv1312.3307C.^ Morimoto, Kanji (1995), “There are knots whose tunnel numbers go down under connected sum”, Proceedings of the American Mathematical Society, 123 (11): 3527\u20133532, doi:10.1090\/S0002-9939-1995-1317043-4, JSTOR\u00a02161103, MR\u00a01317043. See in particular Lemma 5.External links[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/invertible-knot-wikipedia\/#breadcrumbitem","name":"Invertible knot – Wikipedia"}}]}]