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The main purpose of this paper is to show that any embedding of K7 in three-dimensional euclidean space contains a knotted cycle. By a similar but simpler argument, it is also shown that any embedding of K, contains a pair of disjoint cycles which are homologically linked.

Examples suggest that there is a correspondence between L-spaces and 3manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of such manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric 3-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the 2-fold branched covers of non-split alternating links. To do this we prove that the fundamental group of the 2-fold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in Homeo+(S 1 ).While the trivial group obviously satisfies this criterion, in this paper we will adopt the convention that it is not left-orderable.The left-orderability of 3-manifold groups has been studied in work of Boyer, Rolfsen and Wiest [3]. An argument of Howie and Short [18, Lemma 2] shows that the fundamental group of an irreducible 3-manifold M with positive first Betti number is locally indicable, hence left-orderable [4]. More generally, such a group is left-orderable if it admits an epimorphism to a left-orderable group [3, Theorem 1.1(1)].The aim of this note is to establish a connection between L-spaces and the left-orderability of their fundamental groups. Given the results we obtain and those obtained elsewhere [6,7,8,40], we formalise a question which has received attention in the recent literature in the following conjecture. Conjecture 3. An irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.It has been asked by Ozsváth and Szabó whether L-spaces can be characterized as those closed, connected 3-manifolds admitting no co-orientable taut foliations. Thus, in the context of Conjecture 3 it is interesting to consider the following open questions: Does the existence of a co-orientable taut foliation on an irreducible rational homology 3-sphere imply the manifold has a left-orderable fundamental group? Are the two conditions equivalent? Calegari and Dunfield have shown that the existence of a co-orientable taut foliation on an irreducible atoroidal rational homology 3-sphere Y implies that π 1 (Y ) has a left-orderable finite index subgroup [5, Corollary 7.6]. Of course, an affirmative answer to Conjecture 3, combined with [31, Theorem 1.4], would prove that the existence of a co-orientable taut foliation implies left-orderable fundamental group.Our first result verifies the conjecture in the case of Seifert fibred spaces.Theorem 4. Suppose Y is a closed, connected, Seifert fibred 3-manifold. Then Y is an L-space if and only if π 1 (Y ) is not left-orderable.The proof of this theorem in the case where the base orbifold of Y is orientable depends on results of Boyer, Rolfsen ...

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Two (smooth or PL) knots K, K' in S 3 are equivalent if there exists a homeomorphism h: S 3 -• S 3 such that h(K) = K'. This implies that their complements S 3 -K and S 3 -K' are homeomorphic. Here we announce the converse implication.

Let M be a compact, connected, orientable, irreducible 3-manifold such that dM is a torus. An isotopy class c of unoriented simple closed curves in dM will be called a slope. A closed 3-manifold M(c) may be constructed by attaching a solid torus / to M so that c bounds a disk in J.If c and d are two slopes, we denote their (minimal) geometric intersection number by A(c, d). This result is sharp; Fintushel-Stern and Berge have given examples of hyperbolic knots in S 3 for which two Dehn surgeries give lens spaces. In the statements of the following corollaries we use rational numbers as in [R] to parametrize the nontrivial Dehn surgeries on a knot K in S 3 . We will denote by K(r) the result of r-surgery on K.