2-bridge knot

In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points.

Equivalently, these are the knots with bridge number 2, the smallest possible bridge number for a nontrivial knot.

2-bridge links are defined similarly as above, but each component will have one min and max.

2-bridge knots were classified by Horst Schubert, using the fact that the 2-sheeted branched cover of the 3-sphere over the knot is a lens space.

This definition can be used to give a bijection between the set of 2-bridge links and the set of rational numbers; the rational number associated to a given link is called the Schubert normal form of the link (as this invariant was first defined by Schubert[1]), and is precisely the fraction associated to the rational tangle whose numerator closure gives the link.