 | This article may be in need of reorganization to comply with Wikipedia's layout guidelines. (April 2024) |
&sq=&lang=&file=File:Pretzel_knot.svg)
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 – 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.
![{\displaystyle \mathbf {R} ^{2}\times [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b543333db3f8648a3d346a70d258e7af2de16c5)
(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.