. G T ] 7 A ug 2 00 8 Corrigendum to “ Knot Floer homology detects fibred knots ”

We correct a mistake on the citation of JSJ theory in [4]. Some arguments in [4] are also slightly modified accordingly. An important step in [4] uses JSJ theory [2, 3] to deduce some topological information about the knot complement when the knot Floer homology is monic, see [4, Section 6]. The version of JSJ theory cited there is from [1]. However, as pointed out by Kronheimer, the definition of “product regions” in [1] is not the one we want. In this note, we will provide the necessary background on JSJ theory following [2]. Some arguments in [4] will then be modified. Definition 1. An n–manifold pair is a pair (M,T ) where M is an n–manifold and T is an (n − 1)–manifold contained in ∂M . A 3–manifold pair (M,T ) is irreducible ifM is irreducible and T is incompressible. An irreducible 3–manifold pair (M,T ) is Haken if each component ofM contains an incompressible surface. Definition 2. [2, Page 10] A compact 3–manifold pair (S, T ) is called an I–pair if S is an I–bundle over a compact surface, and T is the corresponding ∂I– bundle. A compact 3–manifold pair (S, T ) is called an S–pair if S is a Seifert fibred manifold and T is a union of Seifert fibres in some Seifert fibration. A Seifert pair is a compact 3–manifold pair (S, T ), each component of which is an I–pair or an S–pair. Definition 3. [2, Page 138] A characteristic pair for a compact, irreducible 3– manifold pair (M,T ) is a perfectly-embedded Seifert pair (Σ,Φ) ⊂ (M, int(T )) such that if f is any essential, nondegenerate map of an arbitrary Seifert pair (S, T ) into (M,T ), f is homotopic, as a map of pairs, to a map f ′ such that f (S) ⊂ Σ and f (T ) ⊂ Φ. The definition of a perfectly-embedded pair can be found in [2, Page 4]. We note that the definition requires that Σ∩∂M = Φ, so Σ is disjoint from ∂M−T . The main result in JSJ theory is the following theorem.