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Lens spaces obtainable by surgery on doubly primitive knots. (English) Zbl 1135.57003

In an unpublished paper [Some knots with surgeries yielding lens spaces], J. Berge studied Dehn surgeries on knots in the \(3\)-sphere \(S^3\) yielding lens spaces. He defined a doubly primitive knot as a knot \(K\) contained in a Heegaard splitting surface \(H\) of genus two in \(S^3\) such that \(K\) represents a free generator of the fundamental groups of both of the handlebodies bounded by \(H\). Set \(E(K)=S^3 -\operatorname{int}N(K)\), the exterior of \(K\). He showed that the Dehn surgery on \(K\) with \(\partial (H \cap E(K))\) being the surgery slope yields a lens space \(M\), and that the core circle \(K^*\) of the filled solid torus in \(M\) is a \(1\)-bridge braid, that is, there is a Heegaard splitting torus \(T\) of the \((p,q)\)-lens space \(M\) such that the two solid tori bounded by \(T\) contain meridian disks \(D\), \(D'\) mutually intersecting in \(p\) points, containing \(K^*\) in their union and each intersecting \(K^*\) in a single arc. For the details see [T. Saito, Topology Appl. 154, 1502–1515 (2007; Zbl 1115.57005)].
The paper under review gives an algorithm to decide whether a given lens space is obtainable by Berge’s surgery or not. As an application, it is shown that such surgeries yielding lens spaces with Klein bottles are on \((\pm 5, 3)\)- or \((\pm 7,3)\)-torus knots in \(S^3\), and the resulting lens spaces are of type \((16, 7)\) or \((20, 9)\). We shortly describe the algorithm. A \(1\)-bridge braid \(K^*\) in \(M\) is denoted by \(K(L(p,q),u)\) if there are \(u-1\) points of \(\partial D \cap \partial D'\) on \(\partial D\) between the two points \(K^* \cap T\). Consider the two-bridge link of type \((p,q)\) in \(\lq\lq\)pillowcase with ears” form, take a rational tangle of type \(\infty\) composed of a subarc of an ear and a subarc containing the \(u\)-th \(\lq\lq\)wedge” from the left-bottom side of the pillowcase and substitute the rational tangle of type \(0\) and \(1\) for it to obtain two \(3\)-bridge diagrams. Precisely one of them represents a knot (not a link), and it is the unknot if and only if Berge’s Dehn surgery on \(K(L(p,q),u)\) yields \(S^3\). This follows from the so-called J. A. Montesinos trick [Knots, Groups, 3-Manif.; Pap. dedic. Mem. R.H. Fox, 227–259 (1975; Zbl 0325.55004)].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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References:

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