In the mathematical theory of knots, a pretzel link is a special kind of link. A pretzel link which is also a knot (i.e. a link with one component) is a pretzel knot.
In the standard projection of the pretzel link, there are lefthanded 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.
YouTube Encyclopedic

1/4Views:1 240 5412 3986 059210 591

Poincaré Conjecture  Numberphile

Trefoil knot

Quantum Money from Knots

Murray GellMann: Beauty and truth in physics
Transcription
Contents
Some basic results
The pretzel link is a knot iff both and all the are odd or exactly one of the is even.^{[2]}
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.^{[2]}
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 (1, 1, 1) pretzel knot is the (righthanded) trefoil; the (−1, −1, −1) pretzel knot is its mirror image.
The (5, −1, −1) pretzel knot is the stevedore knot (6_{1}).
If p, q, r are distinct odd integers greater than 1, then the (p, q, r) pretzel knot is a noninvertible 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 .^{[3]}
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.^{[4]}
Utility
(−2, 3, 2n + 1) pretzel links are especially useful in the study of 3manifolds. 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 twocusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the Whitehead link.^{[5]}
References
 ^ "10 124", The Knot Atlas. Accessed November 19, 2017.
 ^ ^{a} ^{b} Kawauchi, Akio (1996). A survey of knot theory. Birkhäuser. ISBN 3764351241
 ^ Zieschang, Heiner (1984), "Classification of Montesinos knots", Topology (Leningrad, 1982), Lecture Notes in Mathematics, 1060, Berlin: Springer, pp. 378–389, doi:10.1007/BFb0099953, MR 0770257
 ^ Montesinos, José M. (1973), "Seifert manifolds that are ramified twosheeted cyclic coverings", Boletín de la Sociedad Matemática Mexicana, 2, 18: 1–32, MR 0341467
 ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2cusped 3manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002993910103645, MR 2661571.
Further reading
 Trotter, Hale F.: Noninvertible knots exist, Topology, 2 (1963), 272–280.
 Burde, Gerhard; Zieschang, Heiner (2003). Knots. De Gruyter studies in mathematics. 5 (2nd revised and extended ed.). Walter de Gruyter. ISBN 3110170051. ISSN 01790986. Zbl 1009.57003.