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.
Schematic picture of a 2-bridge knot.
Bridge number 2
31
51
63
71...
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.
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[1]), and is precisely the fraction associated to the rational tangle whose numerator closure gives the link.[2]: chapter 10
Further readingedit
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. ISBN0-8218-3678-1
Referencesedit
^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. ISBN978-1-4704-5499-9.
bridge, knot, mathematical, field, knot, theory, knot, which, regular, isotoped, that, natural, height, function, given, coordinate, only, maxima, minima, critical, points, equivalently, these, knots, with, bridge, number, smallest, possible, bridge, number, n. 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 Schematic picture of a 2 bridge knot Bridge number 231516371 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 Contents 1 Schubert normal form 2 Further reading 3 References 4 External linksSchubert normal form editThe 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 1 and is precisely the fraction associated to the rational tangle whose numerator closure gives the link 2 chapter 10 Further reading editLouis H Kauffman Sofia Lambropoulou On the classification of rational knots L Enseignement Mathematique 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 1References edit Schubert Horst 1956 Knoten mit zwei Brucken 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 editTable and invariants of rational knots with up to 16 crossings nbsp This knot theory related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title 2 bridge knot amp oldid 1189998163, wikipedia, wiki, book, books, library,
