Tangle (mathematics)

In mathematics, a tangle is generally 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 ${\displaystyle \mathbf {R} ^{2}\times [0,1]}$ – 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.

A third, quite different use of tangle—this one graph theoretical—was introduced by Neil Robertson and Paul Seymour, who use it to describe separation in graphs. This usage has been extended to matroids.

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, strings whose ends are nailed down are used. See also braid theory.