Tangle
A region in a Knot or Link projection plane surrounded by a Circle such that the Knot orLink crosses the circle exactly four times. Two tangles are equivalent if a sequence of Reidemeister Movescan be used to transform one into the other while keeping the four string endpoints fixed and not allowing strings to passoutside the Circle.
The simplest tangles are the 
-tangle and 0-tangle, shown above. A tangle with 
left-handed twists is called an-tangle, and one with right-handed twists is called a 
-tangle. By placing tangles side by side, morecomplicated tangles can be built up such as (
, 3, 2), etc. The link created by connecting the ends of the tangles isnow described by the sequence of tangle symbols, known as Conway's Knot Notation. If tangles are multiplied by 0and then added, the resulting tangle symbols are separated by commas. Additional symbols which are used are the period,colon, and asterisk.
Amazingly enough, two tangles described in this Notation are equivalent Iff the Continued Fractions of the form
are equal (Burde and Zieschang 1985)! An Algebraic Tangle is any tangle obtained by Additionsand Multiplications of rational tangles (Adams 1994). Not all tangles areAlgebraic.See also Algebraic Link, Flype, Pretzel Knot
References
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman pp. 41-51, 1994. Burde, G. and Zieschang, H. Knots. Berlin: de Gruyter, 1985.
- Paradox
- Paradromic Rings
- Paragyrate Diminished Rhombicosidodecahedron
- Parallel
- Parallel Axiom
- Parallel Class
- Parallel Curve
- Parallelepiped
- Parallelism
- Parallelizable
- Parallelogram
- Parallelogram Illusion
- Parallelogram Law
- Parallelohedron
- Parallelotope
- Parallel Postulate
- Parallel (Surface of Revolution)
- Paralogic Triangles
- Parameter
- Parameterization
- Parameter (Quadric)
- Parametric Latitude
- Parametric Test
- Pareto Distribution
- Parity