[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/chiral-knot-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/chiral-knot-wikipedia\/","headline":"Chiral knot – Wikipedia","name":"Chiral knot – Wikipedia","description":"From Wikipedia, the free encyclopedia Knot that is not equivalent to its mirror image In the mathematical field of knot","datePublished":"2019-01-15","dateModified":"2019-01-15","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\/f\/f8\/TrefoilKnot-02.png\/80px-TrefoilKnot-02.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/f8\/TrefoilKnot-02.png\/80px-TrefoilKnot-02.png","height":"80","width":"80"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/chiral-knot-wikipedia\/","wordCount":3293,"articleBody":"From Wikipedia, the free encyclopediaKnot that is not equivalent to its mirror imageIn the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot’s chirality can be further classified depending on whether or not it is invertible.There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.[1]Table of ContentsBackground[edit]Invertible knot[edit]Fully chiral knot[edit]Amphicheiral knot[edit]Fully amphicheiral[edit]Positive amphicheiral[edit]Negative amphicheiral[edit]References[edit]Background[edit]The possible chirality of certain knots was suspected since 1847 when Johann Listing asserted that the trefoil was chiral,[2] and this was proven by Max Dehn in 1914. P. G. Tait found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had even crossing number. Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphicheiral knots in the late 1910s.[3][4] But a counterexample to Tait’s conjecture, a 15-crossing amphicheiral knot, was found by Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks in 1998.[5] However, Tait’s conjecture was proven true for prime, alternating knots.[6]Number of knots of each type of chirality for each crossing numberNumber of crossings345678910111213141516OEIS sequenceChiral knots10227164915255221189988466982532921387166N\/AInvertible knots1022716471253651015306988132671278717A051769Fully chiral knots00000022718711036919378852265801308449A051766Amphicheiral knots010105013058027411539A052401Positive Amphicheiral Noninvertible knots000000000106065A051767Negative Amphicheiral Noninvertible  knots00000106040022711361A051768Fully Amphicheiral knots010104070170410113A052400Both possible trefoil knots.The left-handed trefoil knot.The right-handed trefoil knot.The simplest chiral knot is the trefoil knot, which was shown to be chiral by Max Dehn. All nontrivial torus knots are chiral. The Alexander polynomial cannot distinguish a knot from its mirror image, but the Jones polynomial can in some cases; if Vk(q) \u2260\u00a0Vk(q\u22121), then the knot is chiral, however the converse is not true. The HOMFLY polynomial is even better at detecting chirality, but there is no known polynomial knot invariant that can fully detect chirality.[7]Invertible knot[edit]A chiral knot that can be smoothly deformed to itself with the opposite orientation is classified as a invertible knot.[8] Examples include the trefoil knot.Fully chiral knot[edit]If a knot is not equivalent to its inverse or its mirror image, it is a fully chiral knot, for example the 9 32 knot.[8]Amphicheiral knot[edit]  An amphicheiral knot is one which has an orientation-reversing self-homeomorphism of the 3-sphere, \u03b1, fixing the knot set-wise.All amphicheiral alternating knots have even crossing number. The first amphicheiral knot with odd crossing number is a 15-crossing knot discovered by Hoste et al.[6]Fully amphicheiral[edit]If a knot is isotopic to both its reverse and its mirror image, it is fully amphicheiral. The simplest knot with this property is the figure-eight knot.Positive amphicheiral[edit]If the self-homeomorphism, \u03b1, preserves the orientation of the knot, it is said to be positive amphicheiral. This is equivalent to the knot being isotopic to its mirror.  No knots with crossing number smaller than twelve are positive amphicheiral and noninvertible .[8]Negative amphicheiral[edit]  The first negative amphicheiral knot.If the self-homeomorphism, \u03b1, reverses the orientation of the knot, it is said to be negative amphicheiral.  This is equivalent to the knot being isotopic to the reverse of its mirror image. The noninvertible knot with this property that has the fewest crossings is the knot 817.[8]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, S2CID\u00a018027155, archived from the original (PDF) on 2013-12-15.^ Przytycki, J\u00f3zef H. (1998). “Classical Roots of Knot Theory”. Chaos, Solitons and Fractals. 9 (4\/5): 531\u201345. Bibcode:1998CSF…..9..531P. doi:10.1016\/S0960-0779(97)00107-0.^ Haseman, Mary Gertrude (1918). “XI.\u2014On Knots, with a Census of the Amphicheirals with Twelve Crossings”. Trans. R. Soc. Edinb. 52 (1): 235\u201355. doi:10.1017\/S0080456800012102. S2CID\u00a0123957148.^ Haseman, Mary Gertrude (1920). “XXIII.\u2014Amphicheiral Knots”. Trans. R. Soc. Edinb. 52 (3): 597\u2013602. doi:10.1017\/S0080456800004476. S2CID\u00a0124014620.^ Hoste, Jim; Thistlethwaite, Morwen; Weeks, Jeff (1998). “The First 1,701,936 Knots”. Math. Intell. 20 (4): 33\u201348. doi:10.1007\/BF03025227. S2CID\u00a018027155.^ a b Weisstein, Eric W. “Amphichiral Knot”. MathWorld. Accessed: May 5, 2013.^ Ramadevi, P.; Govindarajan, T.R.; Kaul, R.K. (1994). “Chirality of Knots 942 and 1071 and Chern-Simons Theory”“. Mod. Phys. Lett. A. 9 (34): 3205\u201318. arXiv:hep-th\/9401095. Bibcode:1994MPLA….9.3205R. doi:10.1142\/S0217732394003026. S2CID\u00a0119143024.^ a b c d “Three Dimensional Invariants“, The Knot Atlas.  "},{"@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\/chiral-knot-wikipedia\/#breadcrumbitem","name":"Chiral knot – Wikipedia"}}]}]