# Tangle (mathematics)

Tangle (mathematics)

In knot theory, a branch of mathematics, a pretzel link is a special kind of link. A pretzel link which is also a knot (i.e. a link with one component) is a pretzel knot.

In mathematics, a tangle can mean one of two related concepts:

• In John Conway's definition, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2n marked points on the ball's boundary.
• In link theory, a tangle is an embedding of n arcs and m circles into – 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.

The balance of this article discusses Conway's sense of tangles; for the link theory sense, see that article.

Two n-tangles are considered equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the 3-ball fixed. Tangle theory can be considered analogous to knot theory except instead of closed loops we use strings whose ends are nailed down. See also braid theory.

Read more about Tangle (mathematics):  Tangle Diagrams, Rational and Algebraic Tangles, Conway Notation, Applications

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