Pretzel link
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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 <math>(p_1,p_2,\dots,p_n)</math> pretzel link, there are <math>p_1</math> left-handed crossings in the first tangle, <math>p_2</math> in the second, and, in general, <math>p_n</math> in the nth.
A pretzel link can also be described as a Montesinos link with integer tangles.
[edit] Some basic results
The <math>(p_1,p_2,\dots,p_n)</math> pretzel link is split if at least two of the <math>p_i</math> are zero; but the converse is false.
The <math>(-p_1,-p_2,\dots,-p_n)</math> pretzel link is the mirror image of the <math>(p_1,p_2,\dots,p_n)</math> pretzel link.
The <math>(p_1,p_2,\dots,p_n)</math> pretzel link is link-equivalent (i.e. homotopy-equivalent in <math>S^3</math>) to the <math>(p_2,p_3,\dots,p_n,p_1)</math> pretzel link. Thus, too, the <math>(p_1,p_2,\dots,p_n)</math> pretzel link is link-equivalent to the <math>(p_k,p_{k+1},\dots,p_n,p_1,p_2,\dots,p_{k-1})</math> pretzel link.
The <math>(p_1,p_2,\dots,p_n)</math> pretzel link is link-equivalent to the <math>(p_n,p_{n-1},\dots,p_2,p_1)</math> pretzel link. However, if one orients the links in a canonical way, then these two links have opposite orientations.
[edit] Some examples
The <math>(-1,-1,-1)</math> pretzel knot is the trefoil; the <math>(0,3,-1)</math> pretzel knot is its mirror image.
The <math>(2p,2q,2r)</math> pretzel link is a link formed by three linked unknots.
The <math>(-3,0,-3)</math> pretzel knot is the connected sum of two trefoil knots.
The <math>(0,q,0)</math> pretzel link is the split union of an unknot and another knot.
[edit] Utility
<math>(-2,3,2n+1)</math> 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 <math>(-2,3,7)</math> pretzel knot in particular.
