Tangles

Tangles are mathematical objects used in knot theory to study how knots and links can be built from simpler pieces. Informally, a tangle consists of portions of a knot diagram contained within a disk, with strands entering and exiting the disk through its boundary. The boundary intersections are a fixed finite set of points, most commonly four, which yields a 2-tangle when considered inside a disk with four boundary points. Two tangles are considered equivalent if they can be transformed into one another by ambient isotopy that preserves the boundary.

A tangle can be combined with other tangles to form more complex knots and links. Operations such as tangle sum or concatenation allow researchers to assemble large knots from smaller, well-understood pieces. This modular approach facilitates the classification and construction of knot families.

Rational tangles form a prominent special class. They can be described by sequences of twists and classified by a continued fraction, giving a correspondence with rational numbers. This idea is central to Conway notation and to describing families of knots and links, including Montesinos links, which arise from sums of rational tangles.

More generally, a tangle can be defined as a pair (B, T), where B is a 3-ball and T is a properly embedded one-dimensional set (a collection of arcs and loops) meeting the boundary of B in a fixed finite set of points. This broader perspective encompasses various configurations used in both theoretical investigations and practical applications.

Outside pure mathematics, the term tangle also refers to any entangled mass or knot, such as tangled hair, rope, or wires. In biology, tangle diagrams have been used to model DNA knotting and recombination, illustrating the cross-disciplinary influence of the concept.