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Abstract. Suppose a manifold is produced by finite Dehn surgery on a non-torus alternating knot for which Seifert's algorithm produces a checkerboard surface. By demonstrating that it contains an essential lamination, we prove that such a manifold has R 3 as universal cover and, consequently, is irreducible and has infinite fundamental group. Together with previous work of Roberts, who proved this result in the case of alternating knots for which Seifert's algorithm does not produce a checkerboard surface, and Moser, who classified the manifolds produced by surgery on torus knots, this paper completes the proof that alternating knots satisfy Strong Property P.Mathematics Subject Classification (1991). Primary 57M25; Secondary 57R30.

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A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose M is an oriented 3-manifold with connected boundary a torus, and suppose M contains a properly embedded, compact, oriented, surface R with a single boundary component that is Thurston norm minimizing in H 2 (M, ∂M ). We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if R admits a double-diamond taut sutured manifold decomposition, then every boundary slope except one is strongly realized by a co-oriented taut foliation; that is, the foliation intersects ∂M transversely in a foliation by curves of that slope. In the case that M is the complement of a knot κ in S 3 , the exceptional filling is the meridional one, and hence κ is persistently foliar, by which we mean that every non-trivial slope is strongly realized; hence, restricting attention to rational slopes, every manifold obtained by non-trivial Dehn surgery along κ is foliar. In particular, if R is a Murasugi sum of surfaces R 1 and R 2 , where R 2 is an unknotted band with an even number 2m ≥ 4 of half-twists, then κ = ∂R is persistently foliar. Definition 1.2. A group G is left-orderable if its elements can be given a strict total ordering < which is left invariant, meaning that g < h implies f g < f h for all f, g, h ∈ G.

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