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Let $M$ be a closed connected orientable 3-manifold. Then Kneser tells us that there is a decomposition $M = P_1 \sharp \cdots \sharp P_k$ of $M$ into prime manifolds. Milnor tells us that if $M = Q_1 \sharp \cdots \sharp Q_l$ then $k = l$ and the $Q_i$ are homeomorphic to the $P_j$ in some order.

Is there any theorem that tells us to what extent the spheres that give a prime decomposition are unique? They are not unique up to isotopy at all bit I was wondering if there was some set of moves for getting between any set of prime decomposing spheres.

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    $\begingroup$ I think it would be hard to find better moves than what you get by following along with the proof of the uniqueness of the decomposition - i.e. whenever the proof replaces one sphere with another, that is an allowable move. $\endgroup$
    – Carl
    May 2 '18 at 6:04
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Different sets of decomposing spheres are related by moves called slide homeomorphisms.

Aschenbrenner et al. (2015) provide several references to this statement, see the remark after Theorem 1.2.1 on page 9. Among those, I've found the exposition in McCullough's 1986 paper Mappings of reducible 3-manifolds the most instructive, where the following is proven:

Theorem. Let $M$ be a compact, connected, orientable 3-manifold. Then any orientation-preserving homeomorphism of $M$ is isotopic to a composition of the following four types of homeomorphisms:

  1. homeomorphisms preserving summands,
  2. interchanges of homeomorphic summands,
  3. spins of $S^1\times S^2$ summands, and
  4. slide homeomorphisms.

See pp. 68–69 of [McCullough (1986)] for details regarding these four homeomorphism types, and Section 2.2 in [Zhao (2004)] for a visually aided description of slide homeomorphisms, the ones that are particularly relevant in view of your question.

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