Pretzel link

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 "n"th.

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

ome basic results

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 $S^3$) 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.

ome examples

The $(-1,-1,-1)$ pretzel knot is the trefoil; the $(0,3,-1)$ pretzel knot is its mirror image.

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

$(-2,3,2n+1)$ 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.

References

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