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?