Introduction

A knot is an intentional complication in cordage which may be useful or decorative. Practical knots may be classified as hitches, bends, splices, or knots. A fastens a rope to another object; a unites two rope ends; a is a multi-strand bend or loop.[1] A in the strictest sense serves as a stopper or knob at the end of a rope to keep that end from slipping through a grommet or eye. Knots have excited interest since ancient times for their practical uses, as well as their topological intricacy, studied in the area of mathematics known as knot theory.

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Trefoil

The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic.)

Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve.

The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil.

Tangles

In mathematics, a tangle is generally one of two related concepts:

• In John Conway's definition, an -tangle is a proper embedding of the disjoint union of arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2 marked points on the ball's boundary.
• In link theory, a tangle is an embedding of arcs and circles into {\displaystyle \mathbf {R} ^{2}\times [0,1]} {\mathbf {R}}^{2}\times [0,1] – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance.

An arbitrary tangle diagram of a rational tangle may look very complicated, but there is always a diagram of a particular simple form: start with a tangle diagram consisting of two horizontal (vertical) arcs; add a "twist", i.e. a single crossing by switching the NE and SE endpoints (SW and SE endpoints); continue by adding more twists using either the NE and SE endpoints or the SW and SE endpoints. One can suppose each twist does not change the diagram inside a disc containing previously created crossings.