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.
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 p1 left-handed crossings in the first tangle, p2 in the second, and, in general, pn in the nth.

A pretzel link can also be described as a Montesinos link with integer tangles.

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[edit] Some basic results

The (p_1,p_2,\dots,p_n) pretzel link is split if at least two of the pi 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.

[edit] Some 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.

[edit] 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.

[edit] References

  • Trotter, Hale F.: Non-invertible knots exist, Topology, 2 (1963), 272-280.
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