In the mathematical theory of knots, a pretzellink is a special kind of link. A pretzel link which is also a knot is a pretzel knot. In the standard projection of the pretzel link, there are left-handed crossings in the first tangle, in the second, and, in general, in the nth. A pretzel link can also be described as a Montesinos link with integer tangles.
Some basic results
The pretzel link is a knot iff both and all the are odd or exactly one of the is even. The pretzel link is split if at least two of the are zero; but the converse is false. The pretzel link is the mirror image of the pretzel link. The pretzel link is isotopic to the pretzel link. Thus, too, the pretzel link is isotopic to the pretzel link. The pretzel link is isotopic to the pretzel link. However, if one orients the links in a canonical way, then these two links have opposite orientations.
Some examples
The pretzel knot is the trefoil; the pretzel knot is its mirror image. The pretzel knot is the stevedore knot . If p, q, r are distinct odd integers greater than 1, then the pretzel knot is a non-invertible knot. The pretzel link is a link formed by three linked unknots. The pretzel knot is the connected sum of two trefoil knots. The 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 Montesinos link which is also a knot is a Montesinos knot. A Montesinos link is composed of several rational tangles. One notation for a Montesinos link is. In this notation, and all the and are integers. The Montesinos link given by this notation consists of the sum of the rational tangles given by the integer and the rational tangles These knots and links are named after the Spanish topologist José María Montesinos Amilibia, who first introduced them in 1973.
Utility
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 pretzel knot in particular. The hyperbolic volume of the complement of the pretzel link is 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.
