# gt.geometric topology – Is there a generalized property P – what can we say about framed link descriptions of \$ S ^ 3 \$?

A knot $$K$$ Supposedly Property P, if any non-trivial stretching operation is pending $$K$$ gives a 3-manifold, which is not simply connected. It is known that every node, with the exception of the unknot, has the property P. I wonder what can be said about a connection that allows for a non-trivial stretching operation that yields $$S ^ 3$$,

On $$n$$Components link $$L$$ should have property R when a stretch operation is pending $$L$$ yield $$sharp ^ n S ^ 1 times S ^ 2$$, The generalized Property-R conjecture states that such a link is used in conjunction with the framing $$sharp ^ n S ^ 1 times S ^ 2$$ must be identical to a decoupling that corresponds to an unlink, with each component having 0. This assumption is true $$n = 1$$ but even for unknown $$n = 2$$ – look here. Note that Kirby's theorem describes any two framed links that describe $$sharp ^ n S ^ 1 times S ^ 2$$ must differ by handles along with blowups and blowdowns – generalized Property R claims that the latter movements are not necessary in the case of $$sharp ^ n S ^ 1 times S ^ 2$$ for every couple $$n$$-Komponentenbeschreibungen.

Is there any generalized Property-P conjecture? There are certainly many links that will lead to an operation $$S ^ 3$$ – For example, any handlebody diagram for a 4-manifold with no 1- or 3-handles. In fact, there is a presumption that any simply connected, smooth, closed 4-manifold allows such a description of handle belief (note: this implies S4PC). If so, such a 4-manifold would yield such a framed link.

Considering the Hopf connection either with $$(0,0)$$frame or $$(0.1)$$-framing we get two descriptions of $$S ^ 3$$ that's certainly not synonymous with handles – that's why Property P does not generalize like Property R. Maybe there's a barrier $$f (n)$$so that two $$n$$– framed link descriptions from $$S ^ 3$$ at most require $$f (n)$$ Blowups and blowdowns along with handles to get in between?