Pretzel link

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.


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