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

A knot $$K$$ is said to have the property P if every non trivial surgery Dehn on $$K$$ gives a variety of 3 that is not just connected. We know that all nodes, except unknowns, have the property P. I wonder what can be said of a link that admits a non-trivial Dehn operation that gives $$S ^ 3$$.

A $$n$$-composer link $$L$$ is said to have the property R if a Dehn surgery on $$L$$ returns $$sharp ^ n S ^ 1 times S ^ 2$$. The generalized conjecture of the property R indicates that such a link, as well as the framing used to obtain $$sharp ^ n S ^ 1 times S ^ 2$$ must be equivalent to a non-link with each component having frame 0. This conjecture is true for $$n = 1$$ but unknown even for $$n = 2$$ – see here. Note that by Kirby's theorem, two framed links describing $$sharp ^ n S ^ 1 times S ^ 2$$ differ by slips and explosions and explosions – the generalized property R states that these latter movements are not necessary in the case of $$sharp ^ n S ^ 1 times S ^ 2$$ for any pair of $$n$$-component descriptions.

Is there any kind of generalized conjecture of the P property? There are certainly a lot of links that have a surgery that gives $$S ^ 3$$ for example, any handle body diagram for a collector 4 without 1 or 3 handles. In fact, there is an assumption that any closed simple connected smooth variety 4 admits such a control body description (note: this implies S4PC) – if true, any such variety 4 would produce such a framed link.

Considering the Hopf link with either $$(0,0)$$-frame or $$(0.1)$$-framing, we get two descriptions of $$S ^ 3$$ that the property P does not generalize naively like the property R. There may be a limit. $$f (n)$$, so that two $$n$$-component links framed descriptions of $$S ^ 3$$ need at most $$f (n)$$ explosions and purges, as well as slips to get in between them?