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?