2-bridge knot
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.[1]
Other names for 2-bridge knots are rational knots, 4-plats, and Viergeflechte (German for '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
[edit]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 Schubert[2]), and is precisely the fraction associated to the rational tangle whose numerator closure gives the link.[3]: chapter 10
Further reading
[edit]- 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. ISBN 0-8218-3678-1
References
[edit]- ↑ De Wit, David (2007). "THE 2-BRIDGE KNOTS OF UP TO 16 CROSSINGS" (PDF). Journal of Knot Theory and Its Ramifications. 16 (08): 997–1019. doi:10.1142/S021821650700566X. ISSN 0218-2165. Retrieved 2025-09-06.
- ↑ Schubert, Horst (1956). "Knoten mit zwei Brücken". Mathematische Zeitschrift. 65: 133–170. doi:10.1007/bf01473875.
- ↑ Purcell, Jessica (2020). Hyperbolic knot theory. American Mathematical Society. ISBN 978-1-4704-5499-9.
External links
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