Slam-dunk

:"For the basketball move, see slam dunk (note the lack of a hyphen)."

In mathematics, particularly low-dimensional topology, the slam-dunk is a particular modification of a given surgery diagram in the 3-sphere for a 3-manifold. The name, but not the move, is due to Timothy Cochran. Let "K" be a component of the link in the diagram and "J" be a component that circles "K" as a meridian. Suppose "K" has integer coefficient "n" and "J" has coefficient a rational number "r". Then we can obtain a new diagram by deleting "J" and changing the coefficient of "K" to "n-1/r". This is the slam-dunk.

The name of the move is suggested by the proof that these diagrams give the same 3-manifold. First, do the surgery on "K", replacing a tubular neighborhood of "K" by another solid torus "T" according to the surgery coefficient "n". Since "J" is a meridian, it can be pushed, or "slam dunked", into "T". Since "n" is an integer, "J" intersects the meridian of "T" once, and so "J" must be isotopic to a longitude of "T". Thus when we now do surgery on "J", we can think of it as replacing "T" by another solid torus. This replacement, as shown by a simple calculation, is given by coefficient "n - 1/r".

The inverse of the slam-dunk can be used to change any rational surgery diagram into an integer one, i.e. a surgery diagram on a framed link.

References

* Robert Gompf and Andras Stipsicz, "4-Manifolds and Kirby Calculus", (1999) (Volume 20 in "Graduate Studies in Mathematics"), American Mathematical Society, Providence, RI ISBN 0-8218-0994-6