Dan has very politely pointed out that I've been talking through my hat again. To start with, there are unlinked cicles in 3-space which are not separated by a plane (prime), which demolishes my proposed "criterion". Secondly, the line in Dan's criterion is the meet of the planes in which the respective circles lie --- what I had in mind as a decision "algorithm". Thirdly, in general a k-sphere and l-sphere lie in a (k+1)-flat and (l+1)-flat, which again meet in a line in (k+l+1)-space: there's no need for any induction. Fred Lunnon On 10/20/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
On 10/20/09, Dan Asimov <dasimov@earthlink.net> wrote:
...
Yes. In fact, for disjoint round spheres S^p and S^q in R^n where p+q+1 = n, they link exactly when there exists a line L in R^n that intersects each sphere twice in alternating order.
(In other words, L intersects the spheres in two linked 0-spheres.)
--Dan
Completely different from the criterion I had in mind --- that they are not linked just when there exists a prime disjoint from and separating them!
And Dan's criterion does yield an algorithm --- the line in question simply joins their centres. WFL