Pretzel link

Some basic results

The ( p 1 , p 2 , … , p n ) pretzel link is a knot iff both n and all the p i are odd or exactly one of the p i is even.

The ( p 1 , p 2 , … , p n ) pretzel link is split if at least two of the p i are zero; but the converse is false.

The ( − p 1 , − p 2 , … , − p n ) pretzel link is the mirror image of the ( p 1 , p 2 , … , p n ) pretzel link.

The ( p 1 , p 2 , … , p n ) pretzel link is link-equivalent (i.e. homotopy-equivalent in S³) to the ( p 2 , p 3 , … , p n , p 1 ) pretzel link. Thus, too, the ( p 1 , p 2 , … , p n ) pretzel link is link-equivalent to the ( p k , p k + 1 , … , p n , p 1 , p 2 , … , p k − 1 ) pretzel link.

The ( p 1 , p 2 , … , p n ) pretzel link is link-equivalent to the ( p n , p n − 1 , … , p 2 , p 1 ) pretzel link. However, if one orients the links in a canonical way, then these two links have opposite orientations.

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 K ( e ; α 1 / β 1 , α 2 / β 2 , … , α n / β n ) .

In this notation, e and all the α i and β i are integers. The Montesinos link given by this notation consists of the sum of the rational tangles given by the integer e and the rational tangles α 1 / β 1 , α 2 / β 2 , … , α n / β n

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.