For the basketball move, see slam dunk

Slam dunk

A slam dunk is a type of basketball shot that is performed when a player jumps in the air and manually powers the ball downward through the basket with one or both hands over the rim. This is considered a normal field goal attempt; if successful it is worth two points. The term "slam dunk" was...

 (note the lack of a hyphen).

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly low-dimensional topology

Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...

, the slam-dunk is a particular modification of a given surgery diagram

Dehn surgery

In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...

 in the 3-sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

 for a 3-manifold

3-manifold

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...

. The name, but not the move, is due to Tim 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

Tubular neighborhood

In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve...

 of K by another solid torus

Solid torus

In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...

 T according to the surgery coefficient n. Since J is a meridian, it can be pushed, or "slam dunk

Slam dunk

A slam dunk is a type of basketball shot that is performed when a player jumps in the air and manually powers the ball downward through the basket with one or both hands over the rim. This is considered a normal field goal attempt; if successful it is worth two points. The term "slam dunk" was...

ed", 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.