Pretzel link

Some examples

The (1, 1, 1) pretzel knot is the (right-handed) trefoil; the (−1, −1, −1) pretzel knot is its mirror image.

The (5, −1, −1) pretzel knot is the stevedore knot (6₁).

If p, q, r are distinct odd integers greater than 1, then the (p, q, r) pretzel knot is a non-invertible knot.

The (2p, 2q, 2r) pretzel link is a link formed by three linked unknots.

The (−3, 0, −3) pretzel knot (square knot (mathematics)) is the connected sum of two trefoil knots.

The (0, q, 0) pretzel link is the split union of an unknot and another knot.

Montesinos

A Montesinos link is a special kind of link that generalizes pretzel links (a pretzel link can also be described as a Montesinos link with integer tangles). A Montesinos link which is also a knot (i.e., a link with one component) is a Montesinos knot.

A Montesinos link is composed of several rational tangles. One notation for a Montesinos link is ${\displaystyle K(e;{\alpha }_1/{\beta }_1,{\alpha }_2/{\beta }_2,\dots ,{\alpha }_n/{\beta }_n)}$.

In this notation, ${\displaystyle e}$ and all the ${\displaystyle {\alpha }_i}$ and ${\displaystyle {\beta }_i}$ are integers. The Montesinos link given by this notation consists of the sum of the rational tangles given by the integer ${\displaystyle e}$ and the rational tangles ${\displaystyle {\alpha }_1/{\beta }_1,{\alpha }_2/{\beta }_2,\dots ,{\alpha }_n/{\beta }_n}$

These knots and links are named after the Spanish topologist José María Montesinos Amilibia, who first introduced them in 1973.

Utility

(−2, 3, 2n + 1) pretzel links are especially useful in the study of 3-manifolds. Many results have been stated about the manifolds that result from Dehn surgery on the (−2,3,7) pretzel knot in particular.

The hyperbolic volume of the complement of the (−2,3,8) pretzel link is 4 times Catalan's constant, approximately 3.66. This pretzel link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the Whitehead link.