ART

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.

Conway tangle transformations and operations

The full set of fundamental transformations and operations on 2-tangles, alongside the elementary tangles 0, ∞, ±1 and ±2.

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.

Blue Trefoil Knot

The trefoil knot has Conway notation [3].

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]
See also

Conway knot
Dowker notation
Alexander–Briggs notation

References

"Conway notation", mi.sanu.ac.rs.

"Conway Notation", The Knot Atlas.

Further reading

Conway, J. H. "An Enumeration of Knots and Links, and Some of Their Algebraic Properties." In J. Leech (editor), Computational Problems in Abstract Algebra. Oxford, England. Pergamon Press, pp. 329–358, 1970. pdf available online
Louis H. Kauffman, Sofia Lambropoulou: On the classification of rational tangles. Advances in Applied Mathematics, 33, No. 2 (2004), 199-237. preprint available at arxiv.org.

Knot theory (knots and links)
Hyperbolic

Figure-eight (41) Three-twist (52) Stevedore (61) 62 63 Endless (74) Carrick mat (818) Perko pair (10161) (−2,3,7) pretzel (12n242) Whitehead (52
1) Borromean rings (63
2) L10a140

Satellite

Composite knots
Granny Square Knot sum

Torus

Unknot (01) Trefoil (31) Cinquefoil (51) Septafoil (71) Unlink (02
1) Hopf (22
1) Solomon's (42
1)

Invariants

Alternating Arf invariant Bridge no.
2-bridge Brunnian Chirality
Invertible Crosscap no. Crossing no. Finite type invariant Hyperbolic volume Khovanov homology Genus Knot group Link group Linking no. Polynomial
Alexander Bracket HOMFLY Jones Kauffman Pretzel Prime
list Stick no. Tricolorability Unknotting no. and problem

Notation
and operations

Alexander–Briggs notation Conway notation Dowker notation Flype Mutation Reidemeister move Skein relation Tabulation

Other

Alexander's theorem Berge Braid theory Conway sphere Complement Double torus Fibered Knot List of knots and links Ribbon Slice Sum Tait conjectures Twist Wild Writhe Surgery theory

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License