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The dual knots of doubly primitive knots

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  • Medicine


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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