Pretzel link

In the mathematical theory of knots, 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 mathematical theory of knots, 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 , … , p n ) {displaystyle (p_{1},,p_{2},dots ,,p_{n})} pretzel link, there are p 1 {displaystyle p_{1}} left-handed crossings in the first tangle, p 2 {displaystyle p_{2}} in the second, and, in general, p n {displaystyle p_{n}} in the nth. A pretzel link can also be described as a Montesinos link with integer tangles. The ( p 1 , p 2 , … , p n ) {displaystyle (p_{1},p_{2},dots ,p_{n})} pretzel link is a knot iff both n {displaystyle n} and all the p i {displaystyle p_{i}} are odd or exactly one of the p i {displaystyle p_{i}} is even. The ( p 1 , p 2 , … , p n ) {displaystyle (p_{1},,p_{2},dots ,,p_{n})} pretzel link is split if at least two of the p i {displaystyle p_{i}} are zero; but the converse is false. The ( − p 1 , − p 2 , … , − p n ) {displaystyle (-p_{1},-p_{2},dots ,-p_{n})} pretzel link is the mirror image of the ( p 1 , p 2 , … , p n ) {displaystyle (p_{1},,p_{2},dots ,,p_{n})} pretzel link. The ( p 1 , p 2 , … , p n ) {displaystyle (p_{1},,p_{2},dots ,,p_{n})} pretzel link is isotopic to the ( p 2 , p 3 , … , p n , p 1 ) {displaystyle (p_{2},,p_{3},dots ,,p_{n},,p_{1})} pretzel link. Thus, too, the ( p 1 , p 2 , … , p n ) {displaystyle (p_{1},,p_{2},dots ,,p_{n})} pretzel link is isotopic to the ( p k , p k + 1 , … , p n , p 1 , p 2 , … , p k − 1 ) {displaystyle (p_{k},,p_{k+1},dots ,,p_{n},,p_{1},,p_{2},dots ,,p_{k-1})} pretzel link. The ( p 1 , p 2 , … , p n ) {displaystyle (p_{1},,p_{2},,dots ,,p_{n})} pretzel link is isotopic to the ( p n , p n − 1 , … , p 2 , p 1 ) {displaystyle (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. The (1, 1, 1) pretzel knot is the (right-handed) trefoil; the (−1, −1, −1) pretzel knot is its mirror image.