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.

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.

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 trefoil; the (0, 3, −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 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.

Pretzel knots can be used to introduce students to the essentials of knot theory by making edible pretzels.

References

• Trotter, Hale F.: Non-invertible knots exist, Topology, 2 (1963), 272–280.