Tangle (mathematics) - Rational and Algebraic Tangles

Rational and Algebraic Tangles

A rational tangle is a 2-tangle that is homeomorphic to the trivial 2-tangle as 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.

We can describe such a diagram by considering the numbers given by consecutive twists around the same set of endpoints, e.g. (2, 1, -3) means start with two horizontal arcs, then 2 twists using NE/SE endpoints, then 1 twist using SW/SE endpoints, and then 3 twists using NE/SE endpoints but twisting in the opposite direction from before. The list begins with 0 if you start with two vertical arcs. The diagram with two horizontal arcs is then (0), but we assign (0, 0) to the diagram with vertical arcs. A convention is needed to describe a "positive" or "negative" twist. Often, "rational tangle" refers to a list of numbers representing a simple diagram as described.

The fraction of a rational tangle is then defined as the number given by the continued fraction . The fraction given by (0,0) is defined as . Conway proved that the fraction is well-defined and completely determines the rational tangle up to tangle equivalence. An accessible proof of this fact is given in: . Conway also defined a fraction of an arbitrary tangle by using the Alexander polynomial.

There is an "arithmetic" of tangles with addition, multiplication, and reciprocal operations. An algebraic tangle is obtained from the addition and multiplication of rational tangles.

The numerator closure of a rational tangle is defined as the link obtained by joining the "north" endpoints together and the "south" endpoints also together. The denominator closure is defined similarly by grouping the "east" and "west" endpoints. Rational links are defined to be such closures of rational tangles.

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