In the mathematical field of 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 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 of K by another solid torusT 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.
slam, dunk, this, article, about, mathematics, concept, basketball, move, slam, dunk, other, uses, slam, dunk, disambiguation, mathematical, field, dimensional, topology, slam, dunk, particular, modification, given, surgery, diagram, sphere, manifold, name, mo. This article is about the mathematics concept For the basketball move see Slam dunk For other uses see Slam dunk disambiguation In the mathematical field of 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 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 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 editRobert 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 nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Slam dunk amp oldid 818457501, wikipedia, wiki, book, books, library,
