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In an unpublished paper [Some knots with surgeries yielding lens spaces], J. Berge studied Dehn surgeries on knots in the \(3\)-sphere \(S^3\) yielding lens spaces.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


57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI arXiv


[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
[4] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co. (1985) · Zbl 0568.57001
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.