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.

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 <var>n</var>th.

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.[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.[1]

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 (−1, −1, −1) pretzel knot is the trefoil; the (0, 3, −1) pretzel knot is its mirror image.

The (5, −1, −1) pretzel knot is the stevedore knot (61).

If <var>p</var>, <var>q</var>, <var>r</var> 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 (square knot (mathematics)) is the connected sum of two trefoil knots.

The (0, <var>q</var>, 0) pretzel link is the split union of an unknot and another knot.

Utility

Edible (−2,3,7) pretzel knot

(−2, 3, 2<var>n</var> + 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

  1. ↑ 1.0 1.1 Kawauchi, Akio (1996). A survey of knot theory. Birkhäuser. ISBN 3-7643-5124-1

Further reading

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