Tangle (mathematics)

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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 quite different use of 'tangle' appears in Graph minors X. Obstructions to tree-decomposition by N. Robertson and P. D. Seymour, Journal of Combinatorial Theory B 52 (1991) 153–190, who used 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.