Версия от 11:41, 21 февраля 2024; EducationBot(обсуждение | вклад)(Новая страница: «{{Английская Википедия/Панель перехода}} {{short description|Notation used to describe knots based on operations on tangles}} {{expert needed|mathematics|reason=Description patchwork and in many places incomplete as well|date=November 2008}} File:Conway tangle transformations and operations.svg|thumb|300px|The full set of fundamental transformations and operations on 2-tangles, alongside the elementary tangles 0, ∞, ±1 a...»)
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In knot theory, Conway notation, invented by John Horton Conway, is a way of describing knots that makes many of their properties clear. It composes a knot using certain operations on tangles to construct it.
In Conway notation, the tangles are generally algebraic 2-tangles. This means their tangle diagrams consist of 2 arcs and 4 points on the edge of the diagram; furthermore, they are built up from rational tangles using the Conway operations.
[The following seems to be attempting to describe only integer or 1/n rational tangles]
Tangles consisting only of positive crossings are denoted by the number of crossings, or if there are only negative crossings it is denoted by a negative number. If the arcs are not crossed, or can be transformed into an uncrossed position with the Reidemeister moves, it is called the 0 or ∞ tangle, depending on the orientation of the tangle.
Operations on tangles
If a tangle, a, is reflected on the NW-SE line, it is denoted by −a. (Note that this is different from a tangle with a negative number of crossings.)Tangles have three binary operations, sum, product, and ramification,[1] however all can be explained using tangle addition and negation. The tangle product, a b, is equivalent to −a+b. and ramification or a,b, is equivalent to −a+−b.
Advanced concepts
Rational tangles are equivalent if and only if their fractions are equal. An accessible proof of this fact is given in (Kauffman and Lambropoulou 2004). A number before an asterisk, *, denotes the polyhedron number; multiple asterisks indicate that multiple polyhedra of that number exist.[2]
