Tangle (mathematics)

In mathematics, particularly in the field of knot theory, a tangle is a generalization of a knot or link. Specifically, it is a pair of sets of arcs properly embedded in a 3-dimensional ball, where the endpoints of the arcs lie on the boundary of the ball. More formally:

A tangle (more accurately, a 2-tangle) consists of a 3-dimensional ball B, with four distinguished points N, E, S, and W on its boundary, and two smooth arcs properly embedded in B such that their endpoints are N, E, S, and W. "Properly embedded" means that the arcs' interiors lie in the interior of B, and the arcs intersect the boundary of B only at their endpoints.

The term "tangle" is often used more loosely to refer to n-tangles for any integer n. An n-tangle is a 3-ball with 2n distinguished points on the boundary, and n arcs properly embedded in the ball with their endpoints at the distinguished points.

Tangles are useful building blocks for constructing more complicated knots and links. By joining tangles together along their boundary points, one can create new tangles, knots, or links. This is particularly useful for analyzing the structure of complex knots and decomposing them into simpler components.

Operations on tangles, such as rational tangle replacement, are important tools in knot theory. Rational tangles are a special class of 2-tangles that can be obtained by a sequence of twists and rotations. These are important in understanding surgery on knots and links.

The study of tangles involves examining their topological properties, such as whether they are equivalent under certain types of deformations. Tangle equivalences are typically defined by ambient isotopy that fixes the boundary of the ball. Tangle theory provides a framework for classifying and understanding the intricate structure of knots and links.