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The (−2,3,7) pretzel knot has two right-handed twists in its first tangle, three left-handed twists in its second, and seven left-handed twists in its third.
P(5,3,-2) = T(5,3) = 10124
P(3,3,-2) = T(4,3) = 819
Only two knots are both torus and pretzel[1]

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 ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link, there are ${\displaystyle p_{1}}$ left-handed crossings in the first tangle, ${\displaystyle p_{2}}$ in the second, and, in general, ${\displaystyle p_{n}}$ in the nth.

A pretzel link can also be described as a Montesinos link with integer tangles.

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## Some basic results

The ${\displaystyle (p_{1},p_{2},\dots ,p_{n})}$ pretzel link is a knot iff both ${\displaystyle n}$ and all the ${\displaystyle p_{i}}$ are odd or exactly one of the ${\displaystyle p_{i}}$ is even.[2]

The ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link is split if at least two of the ${\displaystyle p_{i}}$ are zero; but the converse is false.

The ${\displaystyle (-p_{1},-p_{2},\dots ,-p_{n})}$ pretzel link is the mirror image of the ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link.

The ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link is isotopic to the ${\displaystyle (p_{2},\,p_{3},\dots ,\,p_{n},\,p_{1})}$ pretzel link. Thus, too, the ${\displaystyle (p_{1},\,p_{2},\dots ,\,p_{n})}$ pretzel link is isotopic to the ${\displaystyle (p_{k},\,p_{k+1},\dots ,\,p_{n},\,p_{1},\,p_{2},\dots ,\,p_{k-1})}$ pretzel link.[2]

The ${\displaystyle (p_{1},\,p_{2},\,\dots ,\,p_{n})}$ pretzel link is isotopic to the ${\displaystyle (p_{n},\,p_{n-1},\dots ,\,p_{2},\,p_{1})}$ 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 (pqr) pretzel knot is a non-invertible knot.

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. In this example, ${\displaystyle e=-3}$ , ${\displaystyle \alpha _{1}/\beta _{1}=-3/2}$ and ${\displaystyle \alpha _{2}/\beta _{2}=5/2}$ .

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 ${\displaystyle K(e;\alpha _{1}/\beta _{1},\alpha _{2}/\beta _{2},\ldots ,\alpha _{n}/\beta _{n})}$.[3]

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 ${\displaystyle \alpha _{1}/\beta _{1},\alpha _{2}/\beta _{2},\ldots ,\alpha _{n}/\beta _{n}}$

These knots and links are named after the Spanish topologist José María Montesinos Amilibia, who first introduced them in 1973.[4]

## Utility

Edible (−2,3,7) pretzel knot

(−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.[5]

## References

1. ^ "10 124", The Knot Atlas. Accessed November 19, 2017.
2. ^ a b Kawauchi, Akio (1996). A survey of knot theory. Birkhäuser. ISBN 3-7643-5124-1
3. ^ 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
4. ^ Montesinos, José M. (1973), "Seifert manifolds that are ramified two-sheeted cyclic coverings", Boletín de la Sociedad Matemática Mexicana, 2, 18: 1–32, MR 0341467
5. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.