# The dual knots of doubly primitive knots

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Saito, T. Osaka J. Math. 45 (2008), 403–421 THE DUAL KNOTS OF DOUBLY PRIMITIVE KNOTS TOSHIO SAITO (Received October 19, 2006, revised April 4, 2007) Abstract For certain (1, 1)-knots in lens spaces with a longitudinal surgery yielding the 3-sphere, we determine a non-negative integer derived from its (1, 1)-splitting. The value will be an invariant for such knots. Roughly, it corresponds to a ‘minimal’ self-intersection number when one consider projections of a knot on a Heegaard torus. As an application, we give a necessary and sufficient condition for such knots to be hyperbolic. 1. Introduction A lens space L(p, q) is a 3-manifold obtained by the p=q-surgery on a trivial knot in the 3-sphere S3 and is homeomorphic neither to S3 nor to S2� S1. Throughout this paper, �L(p, q) denotes the same manifold as L(p, q) with reversed orientation. A knot K in a closed orientable 3-manifold M is called a (1, 1)-knot if (M , K ) = (V1, t1)[P (V2, t2), where (V1, V2; P) is a genus one Heegaard splitting and ti is a trivial arc in Vi (i = 1 and 2). (An arc t properly embedded in a solid torus V is said to be trivial if there is a disk D in V with t � �D and �Dnt � �V .) Set Wi = (Vi , ti ) (i = 1 and 2). We call the triplet (W1, W2; P) a (1, 1)-splitting of (M , K ). We regard P as a torus with two specified points P \K . Let E1 (E2 resp.) be a meridian disk of V1 (V2 resp.) disjoint from t1 (t2 resp.). It is known that such a disk is unique up to isotopy on V1 n t1 (V2 n t2 resp.) (cf. [13, Lemma 3.4]). A (1, 1)-splitting (W1, W2; P) is said to be monotone if the signed intersection points of �E1 and �E2 have the same sign for some orientations of �E1 and �E2. Berge’s work [1] indicates that it is very important to study (1, 1)-knots. Which knots in S3 admit Dehn surgeries yielding lens spaces? This problem is still open. In [1], Berge introduced the concept of doubly primitive knots and gave an integral surgery to obtain a lens space from any doubly primitive knot. In this paper, we cal

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