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The amalgamation of high distance Heegaard splittings is always efficient. (English) Zbl 1140.57012
Summary: Let \(M\) be a compact orientable 3-manifold, and \(F\) be an essential closed surface which cuts \(M\) into two 3-manifolds \(M_{1}\) and \(M_{2}\). Let \({M_{i}=V_{i}\cup_{S_{i}} W_{i}}\) be a Heegaard splitting for \(i = 1, 2\). We denote by \(d(S_{i})\) the distance of \({V_{i}\cup_{S_{i}} W_{i}}\). If \(d(S_{1}), d(S_{2}) \geq 2(g (M_{1}) + g (M_{2}) - g (F))\), then \(M\) has a unique minimal Heegaard splitting up to isotopy, i.e. the amalgamation of \({V_{1}\cup_{S_{1}} W_{1}}\) and \({V_{2}\cup_{S_{2}} W_{2}}\).

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI
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