Conway notation (knot theory)
{{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
File:Blue Trefoil Knot.png has Conway notation [3].]]
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.
Basic concepts
=Tangles=
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,"[http://www.mi.sanu.ac.rs/vismath/sl/l14.htm Conway notation]", mi.sanu.ac.rs. 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.{{Knot Atlas|Conway Notation}}
See also
References
{{reflist}}
Further reading
- {{cite book |first=J.H. |last=Conway |chapter=An Enumeration of Knots and Links, and Some of Their Algebraic Properties |chapter-url=http://www.maths.ed.ac.uk/~aar/papers/conway.pdf |editor-first=J. |editor-last=Leech |title=Computational Problems in Abstract Algebra |publisher=Pergamon Press |year=1970 |isbn=0080129757 |pages=329–358 |url=}}
- {{cite journal |first1=Louis H. |last1=Kauffman |first2=Sofia |last2=Lambropoulou |title=On the classification of rational tangles |journal=Advances in Applied Mathematics |volume=33 |issue=2 |pages=199–237 |year=2004 |doi=10.1016/j.aam.2003.06.002 |arxiv=math/0311499|s2cid=119143716 }}
{{Knot theory|state=collapsed}}