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It is a famous theorem of Waldhausen that any genus g Heegaard splitting surface H in S³ is isotopic to the standard genus g splitting surface H₀. Rieck’s modern account of Waldhausen's theorem exploits a theorem of Casson and Gordon on weakly reducible Heegaard splittings. They show that a weak reduction of the splitting leads to a way of breaking up the surface into a connected sum of lower genus splittings, at which point induction can be used.

But how unique is the isotopy of H to H₀? To what extent is the isotopy determined by the choice of a weak reduction? We discuss this in the context of a conjecture of Powell about isotopies of H₀ to itself.