##
**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In an unpublished paper [Some knots with surgeries yielding lens spaces], J. Berge studied Dehn surgeries on knots in the \(3\)-sphere \(S^3\) yielding lens spaces.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)].

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)].

Reviewer: Chuichiro Hayashi (Tokyo)

### MSC:

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

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\textit{K. Ichihara} and \textit{T. Saito}, Algebr. Geom. Topol. 7, 1949--1962 (2007; Zbl 1135.57003)

### References:

[1] | J Berge, Some knots with surgeries yielding lens spaces, unpublished manuscript |

[2] | S A Bleiler, R A Litherland, Lens spaces and Dehn surgery, Proc. Amer. Math. Soc. 107 (1989) 1127 · Zbl 0686.57007 |

[3] | G E Bredon, J W Wood, Non-orientable surfaces in orientable 3-manifolds, Invent. Math. 7 (1969) 83 · Zbl 0175.20504 |

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[5] | R Fintushel, R J Stern, Constructing lens spaces by surgery on knots, Math. Z. 175 (1980) 33 · Zbl 0425.57001 |

[6] | T Homma, M Ochiai, On relations of Heegaard diagrams and knots, Math. Sem. Notes Kobe Univ. 6 (1978) 383 · Zbl 0391.57006 |

[7] | R Kirby, Problems in low-dimensional topology (editor R Kirby), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35 |

[8] | J M Montesinos, Surgery on links and double branched covers of \(S^3\) (editor L P Neuwirth), Ann. of Math. Studies 84, Princeton Univ. Press (1975) 227 · Zbl 0325.55004 |

[9] | L Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971) 737 · Zbl 0202.54701 |

[10] | T Saito, The dual knots of doubly primitive knots, to appear in Osaka J. Math. · Zbl 1146.57012 |

[11] | T Saito, Knots in lens spaces with the 3-sphere surgery, preprint · Zbl 1170.57008 |

[12] | T Saito, Dehn surgery and \((1,1)\)-knots in lens spaces, Topology Appl. 154 (2007) 1502 · Zbl 1115.57005 |

[13] | S C Wang, Cyclic surgery on knots, Proc. Amer. Math. Soc. 107 (1989) 1091 · Zbl 0688.57008 |

[14] | Y Q Wu, Cyclic surgery and satellite knots, Topology Appl. 36 (1990) 205 · Zbl 0715.57002 |

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