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Do Heegaard Floer homology detect fibred knot in general oriented 3-manifolds other than $S^3$? If the answer is yes could you give a reference.

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This is known for any knot in a closed, oriented 3-manifold whose complement is irreducible. Ni proved this in Knot Floer homology detects fibred knots, building on Ghiggini's proof in the genus 1 case; another proof was given by Juhász (see theorem 9.11).

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