A rank-minimality condition in sutured Heegaard Floer homology forces a connect-sum decomposition with a Floer simple knot, and link Floer homology detects spherical braid closures in S^1×S^2.
The decategorification of sutured Floer homology.Journal of Topology, 4(2):431–478, 2011
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On Heegaard Floer minimal knots in sutured manifolds
A rank-minimality condition in sutured Heegaard Floer homology forces a connect-sum decomposition with a Floer simple knot, and link Floer homology detects spherical braid closures in S^1×S^2.