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Conway notation (knot theory)

Today, the topic of Conway notation (knot theory) is of great relevance in society. With the advancement of technology and constant changes in the world, Conway notation (knot theory) has become a point of interest for many people. Whether Conway notation (knot theory) is a cultural phenomenon, a scientific discovery, or a historical figure, his impact on our lives is undeniable. In this article, we will explore different aspects of Conway notation (knot theory) and its influence in different areas, providing a broad and enriching vision on this topic that continues to capture the attention of audiences around the world.

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

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.

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]

See also

References

  1. ^ "Conway notation", mi.sanu.ac.rs.
  2. ^ "Conway Notation", The Knot Atlas.

Further reading

  • Conway, J.H. (1970). "An Enumeration of Knots and Links, and Some of Their Algebraic Properties" (PDF). In Leech, J. (ed.). Computational Problems in Abstract Algebra. Pergamon Press. pp. 329–358. ISBN 0080129757.
  • Kauffman, Louis H.; Lambropoulou, Sofia (2004). "On the classification of rational tangles". Advances in Applied Mathematics. 33 (2): 199–237. arXiv:math/0311499. doi:10.1016/j.aam.2003.06.002. S2CID 119143716.
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