Tangle (mathematics)

In mathematics, a tangle is generally one of two related concepts: (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.

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.

Without loss of generality, consider the marked points on the 3-ball boundary to lie on a great circle.

The tangle can be arranged to be in general position with respect to the projection onto the flat disc bounded by the great circle.

A rational tangle is a 2-tangle that is homeomorphic to the trivial 2-tangle by a map of pairs consisting of the 3-ball and two arcs.

The four endpoints of the arcs on the boundary circle of a tangle diagram are usually referred as NE, NW, SW, SE, with the symbols referring to the compass directions.

An arbitrary tangle diagram of a rational tangle may look very complicated, but there is always a diagram of a particular simple form: start with a tangle diagram consisting of two horizontal (vertical) arcs; add a "twist", i.e. a single crossing by switching the NE and SE endpoints (SW and SE endpoints); continue by adding more twists using either the NE and SE endpoints or the SW and SE endpoints.

One can suppose each twist does not change the diagram inside a disc containing previously created crossings.

Often, "rational tangle" refers to a list of numbers representing a simple diagram as described.

[2] Conway also defined a fraction of an arbitrary tangle by using the Alexander polynomial.

The denominator closure is defined similarly by grouping the "east" and "west" endpoints.

One motivation for Conway's study of tangles was to provide a notation for knots more systematic than the traditional enumeration found in tables.

The (−2,3,7) pretzel knot has two right-handed twists in its first tangle , three left-handed twists in its second, and seven left-handed twists in its third.
Some operations on tangles:
Left: A tangle a and its reflection − a . Top right: Tangle addition, denoted by a + b . Center right: Tangle product, denoted by a b , equivalent to − a + b . Bottom right: Ramification, denoted by a , b , equivalent to − a + − b