Pretzel link

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.

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

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

Some basic results

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

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

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

The $(p_{1},\,p_{2},\dots ,\,p_{n})$ pretzel link is link-equivalent (i.e. homotopy-equivalent in S³) to the $(p_{2},\,p_{3},\dots ,\,p_{n},\,p_{1})$ pretzel link. Thus, too, the $(p_{1},\,p_{2},\dots ,\,p_{n})$ pretzel link is link-equivalent to the $(p_{k},\,p_{{k+1}},\dots ,\,p_{n},\,p_{1},\,p_{2},\dots ,\,p_{{k-1}})$ pretzel link.^[1]

The $(p_{1},\,p_{2},\,\dots ,\,p_{n})$ pretzel link is link-equivalent to the $(p_{n},\,p_{{n-1}},\dots ,\,p_{2},\,p_{1})$ 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 (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. 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 ${\displaystyle K(e;\alpha _{1}/\beta _{1},\alpha _{2}/\beta _{2},\ldots ,\alpha _{n}/\beta _{n})}$.^[2]

In this notation, $e$ and all the $\alpha _{i}$ and $\beta _{i}$ 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 ${\displaystyle \alpha _{1}/\beta _{1},\alpha _{2}/\beta _{2},\ldots ,\alpha _{n}/\beta _{n}}$

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.^[3]

References

1. Kawauchi, Akio (1996). A survey of knot theory. Birkhäuser. ISBN 3-7643-5124-1
2. Zieschang, Heiner. "Classification of Montesinos knots." Topology. Springer Berlin Heidelberg, 1984. 378–389
3. Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.

Further reading

• Trotter, Hale F.: Non-invertible 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 0179-0986. Zbl 1009.57003.