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 knot theory, a branch of mathematics, 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.

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.^[2]

References

1. Kawauchi, Akio (1996). A survey of knot theory. Birkhäuser. ISBN 3-7643-5124-1
2. 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.