   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.kawauchi, akio (1996). a survey of knot theory. birkh&auml;user. isbn 3-7643-5124-1 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. 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 (â1,&nbsp;â1,&nbsp;â1) pretzel knot is the trefoil; the (0,&nbsp;3,&nbsp;â1) pretzel knot is its mirror image. the (5,&nbsp;â1,&nbsp;â1) pretzel knot is the stevedore knot&nbsp;(61). if p, q, r are distinct odd integers greater than 1, then the (p,&nbsp;q,&nbsp;r) pretzel knot is a non-invertible knot. the (2p,&nbsp;2q,&nbsp;2r) pretzel link is a link formed by three linked unknots. the (â3,&nbsp;0,&nbsp;â3) pretzel knot (square knot (mathematics)) is the connected sum of two trefoil knots. the (0,&nbsp;q,&nbsp;0) pretzel link is the split union of an unknot and another knot. ==utility== right|thumb|edible (â2,3,7) pretzel knot (â2,&nbsp;3,&nbsp;2n&nbsp;+&nbsp;1) pretzel links are especially useful in the study of 3-manifolds. many results have b
3-manifolds Pretzel_knots_and_links_(mathematics)