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Pretzel link
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{{short description|Link formed from a finite number of twisted sections}} {{refimprove|date=August 2010}} [[Image:Pretzel knot.svg|right|140px|thumb|The [[(−2,3,7) pretzel knot]] has two right-handed twists in its first [[Tangle (mathematics)|tangle]], three left-handed twists in its second, and seven left-handed twists in its third.]] {{multiple image | width = 96 | image1 = A (5,3)-torus knot.png | caption1 = P(5,3,-2) = T(5,3) = 10<sub>124</sub> | image2 = A (4,3)-torus knot.png | caption2 = P(3,3,-2) = T(4,3) = 8<sub>19</sub> | footer = Only two knots are both torus and pretzel<ref>{{Knot Atlas|10 124}} Accessed November 19, 2017.</ref> }} In the [[knot theory|mathematical theory of knots]], a '''pretzel link''' is a special kind of [[link (knot theory)|link]]. It consists of a finite number of [[Tangle (mathematics)|tangle]]s made of two intertwined circular [[helix|helices]]. The tangles are connected cyclicly,<ref>[https://mathcurve.com/courbes3d.gb/bretzel/bretzel.shtml Pretzel link at Mathcurve]</ref> and the first component of the first tangle is connected to the second component of the second tangle, the first component of the second tangle is connected to the second component of the third tangle, and so on. Finally, the first component of the last tangle is connected to the second component of the first. A pretzel link which is also a [[knot (mathematics)|knot]] (that is, a link with one component) is a '''pretzel knot'''. Each tangle is characterized by its number of twists: positive if they are counter-clockwise or left-handed, negative if clockwise or right-handed. In the standard projection of the <math>(p_1,\,p_2,\dots,\,p_n)</math> pretzel link, there are <math>p_1</math>left-handed crossings in the first tangle, <math>p_2</math>in the second, and, in general, <math>p_n</math>in the <var>n</var>th. A pretzel link can also be described as a [[#Montesinos|Montesinos link]] with integer tangles. ==Some basic results== The <math>(p_1,p_2,\dots,p_n)</math> pretzel link is a [[knot (mathematics)|knot]] [[iff]] both <math>n</math> and all the <math>p_i</math> are [[Even and odd numbers|odd]] or exactly one of the <math>p_i</math> is even.<ref name="kawauchi">Kawauchi, Akio (1996). ''A survey of knot theory''. Birkhäuser. {{ISBN|3-7643-5124-1}}</ref> The <math>(p_1,\,p_2,\dots,\,p_n)</math> pretzel link is [[Split link|split]] if at least two of the <math>p_i</math> are [[0 (number)|zero]]; but the [[Converse (logic)|converse]] is false. The <math>(-p_1,-p_2,\dots,-p_n)</math> pretzel link is the [[mirror image]] of the <math>(p_1,\,p_2,\dots,\,p_n)</math> pretzel link. The <math>(p_1,\,p_2,\dots,\,p_n)</math> pretzel link is isotopic to the <math>(p_2,\,p_3,\dots,\,p_n,\,p_1)</math> pretzel link. Thus, too, the <math>(p_1,\,p_2,\dots,\,p_n)</math> pretzel link is isotopic to the <math>(p_k,\,p_{k+1},\dots,\,p_n,\,p_1,\,p_2,\dots,\,p_{k-1})</math> pretzel link.<ref name="kawauchi" /> The <math>(p_1,\,p_2,\,\dots,\,p_n)</math> pretzel link is isotopic to the <math>(p_n,\,p_{n-1},\dots,\,p_2,\,p_1)</math> pretzel link. However, if one orients the links in a canonical way, then these two links have opposite orientations. ==Some examples== The (1, 1, 1) pretzel knot is the (right-handed) [[trefoil knot|trefoil]]; the (−1, −1, −1) pretzel knot is its [[mirror image]]. The (5, −1, −1) pretzel knot is the [[Stevedore knot (mathematics)|stevedore knot]] (6<sub>1</sub>). If <var>p</var>, <var>q</var>, <var>r</var> are distinct odd integers greater than 1, then the (''p'', ''q'', ''r'') pretzel knot is a [[non-invertible knot]]. The (2''p'', 2''q'', 2''r'') pretzel link is a link formed by three linked [[unknot]]s. The (−3, 0, −3) pretzel knot ([[square knot (mathematics)]]) is the [[connected sum]] of two [[trefoil knot]]s. The (0, <var>q</var>, 0) pretzel link is the [[Split link|split union]] of an [[unknot]] and another knot. ===Montesinos=== A '''Montesinos link''' is a special kind of [[link (knot theory)|link]] that generalizes pretzel links (a pretzel link can also be described as a Montesinos link with integer tangles). A Montesinos link which is also a [[knot (mathematics)|knot]] (i.e., a link with one component) is a '''Montesinos knot'''. A Montesinos link is composed of several [[rational tangles]]. One notation for a Montesinos link is <math>K(e;\alpha_1 /\beta_1,\alpha_2 /\beta_2,\ldots,\alpha_n /\beta_n)</math>.<ref>{{citation|last=Zieschang|first=Heiner|authorlink=Heiner Zieschang| contribution=Classification of Montesinos knots|title= Topology (Leningrad, 1982)| series=Lecture Notes in Mathematics| volume=1060|publisher=Springer|location= Berlin|year= 1984| pages=378–389|mr=0770257|doi=10.1007/BFb0099953}}</ref> In this notation, <math>e</math> and all the <math>\alpha_i</math> and <math>\beta_i</math> are integers. The Montesinos link given by this notation consists of the [[Tangle (mathematics)#Operations on tangles|sum]] of the rational tangles given by the integer <math>e</math> and the rational tangles <math>\alpha_1 /\beta_1,\alpha_2 /\beta_2,\ldots,\alpha_n /\beta_n</math> These knots and links are named after the Spanish topologist [[José María Montesinos Amilibia]], who first introduced them in 1973.<ref>{{citation|mr=0341467|last=Montesinos|first= José M.|title= Seifert manifolds that are ramified two-sheeted cyclic coverings| journal=Boletín de la Sociedad Matemática Mexicana|series= 2|volume= 18 |year=1973|pages=1–32}}</ref> ==Utility== (−2, 3, 2<var>n</var> + 1) pretzel links are especially useful in the study of [[3-manifold]]s. Many results have been stated about the manifolds that result from [[Dehn surgery]] on the [[(−2,3,7) pretzel knot]] in particular. The [[hyperbolic volume]] of the complement of the {{math|(−2,3,8)}} pretzel link is {{math|4}} times [[Catalan's constant]], approximately 3.66. This pretzel link complement is one of two two-cusped [[hyperbolic manifold]]s with the minimum possible volume, the other being the complement of the [[Whitehead link]].<ref>{{citation | last = Agol | first = Ian | authorlink = Ian Agol | doi = 10.1090/S0002-9939-10-10364-5 | issue = 10 | journal = [[Proceedings of the American Mathematical Society]] | mr = 2661571 | pages = 3723–3732 | title = The minimal volume orientable hyperbolic 2-cusped 3-manifolds | volume = 138 | year = 2010| arxiv = 0804.0043}}.</ref> {{multiple image | align = center | image1 = MontesinosLink1.svg | width1 = 300 | alt1 = Link diagram showing a Montesinos link | caption1 = A Montesinos link. In this example, <math>e=-3</math> , <math>\alpha_1 /\beta_1=-3/2</math> and <math>\alpha_2 /\beta_2=5/2</math>. | image2 = PretzelKnot.jpg | width2 = 323 | alt2 = A pretzel baked in the shape of a (–2,3,7) pretzel knot | caption2 = Edible (−2,3,7) pretzel knot | image3 = Another (-2,3,7) pretzel knot.jpg | width3 = 240 | alt3 = A pretzel baked in the shape of a (–2,3,7) pretzel knot, with shiny egg glaze | caption3 = Another edible (–2,3,7) pretzel knot, glazed to perfection }} ==References== {{reflist}} ==Further reading== * Trotter, Hale F.: ''Non-invertible knots exist'', '''Topology''', 2 (1963), 272–280. * {{cite book | title=Knots | volume=5 | series=De Gruyter studies in mathematics | issn=0179-0986 | first1=Gerhard | last1=Burde | first2=Heiner | last2=Zieschang | edition=2nd revised and extended | zbl=1009.57003 | publisher=Walter de Gruyter | year=2003 | isbn=3110170051 }} {{Knot theory}} [[Category:Pretzel knots and links (mathematics)| ]] [[Category:3-manifolds]]
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