2-bridge knot

Type of knot in knot theory thumb|Schematic picture of a 2-bridge knot. 2 60 Bridge number 2 2-bridge trefoil.svg 31 Blue Cinquefoil Knot.png 51 Blue 6 3 Knot.png 63 Blue 7 1 Knot.png 71... In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points. Equivalently, these are the knots with bridge number 2, the smallest possible bridge number for a nontrivial knot. Every nontrivial knot with up to seven crossings is a 2-bridge knot. The simplest knots with a bridge number of 3 have eight crossings. Of the 1,701,936 knots with up to sixteen crossings, 5,546 are 2-bridge knots.De Wit David THE 2-BRIDGE KNOTS OF UP TO 16 CROSSINGS Journal of Knot Theory and Its Ramifications 16 08 2007 0218-2165 10.1142/S021821650700566X free 997–1019 https://arxiv.org/pdf/math/0409351v2.pdf 2025-09-06 Other names for 2-bridge knots are rational knots, 4-plats, and de Viergeflechte four braids. 2-bridge links are defined similarly as above, but each component will have one min and max. 2-bridge knots were classified by Horst Schubert, using the fact that the 2-sheeted branched cover of the 3-sphere over the knot is a lens space. Schubert normal form The names rational knot and rational link were coined by John Conway who defined them as arising from numerator closures of rational tangles. This definition can be used to give a bijection between the set of 2-bridge links and the set of rational numbers; the rational number associated to a given link is called the Schubert normal form of the link (as this invariant was first defined by SchubertSchubert Horst Knoten mit zwei Brücken Mathematische Zeitschrift 1956 65 133–170 10.1007/bf01473875), and is precisely the fraction associated to the rational tangle whose numerator closure gives the link.Purcell Jessica Hyperbolic knot theory 2020 American Mathematical Society 978-1-4704-5499-9chapter 10 Further reading Louis H. Kauffman, Sofia Lambropoulou: On the classification of rational knots, L' Enseignement Mathématique, 49:357–410 (2003). preprint available at arxiv.org C. C. Adams, The Knot Book: An elementary introduction to the mathematical theory of knots. American Mathematical Society, Providence, RI, 2004. xiv+307 pp. 0-8218-3678-1 References External links Table and invariants of rational knots with up to 16 crossings Category:Knot theory