Pretzel link

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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.

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.

[edit] Some 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.

[edit] Some examples

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

The $(2 p,2 q,2 r)$ 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.

[edit] Utility

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