##
**On hyperbolic 3-manifolds obtained by Dehn surgery on links.**
*(English)*
Zbl 1210.57018

The authors study Dehn surgeries along a certain family of links in \(S^3\). The links are denoted by \({\mathcal L}_{(m,d)}\) where \(d\) and \(m\) are positive integers. They are described in the second paragraph. The authors discuss hyperbolicity, presentations of the fundamental group, and expression as framed links, of the manifolds obtained by surgery along \({\mathcal L}_{(m,d)}\).

Let \(L_i\)\((i=1, \dots, d)\) be the \(1/m\)-rational tangle (i.e.\((-m)\)-half twists) with \(d\) and \(m\) positive integers. By arranging them linearly, we obtain a pretzel link consisting of \(d\) times \((-m)\)-half twists. Then there is an orientation-preserving autohomeomorphism of \(S^3\) with period \(d\) which fixes the pretzel link setwise, and maps \(L_i\) to \(L_{i+1}\) where \(L_{d+1}=L_1\). Then the link \({\mathcal L}_{(m,d)}\) is a union of the pretzel link and the axis of the periodic map. The link \({\mathcal L}_{(m,d)}\) is also periodic with period \(d\), and is strongly invertible along the axis perpendicular to all \(L_i\). We can also define \({\mathcal L}_{(m,d)}\) for \(m<0\) as the mirror image of \({\mathcal L}_{(-m,d)}\). The number of the components of \({\mathcal L}_{(m,d)}\) is \(2\) if both \(d\) and \(m\) are odd, \(3\) if \(d\) is even and \(m\) is odd, and \(d+1\) if \(m\) is even. There are two special families : (1)\({\mathcal L}_{(m,1)}\) is a \(2\)-bridge link of type \((2, m, -2)\) in Conway’s notation. \({\mathcal L}_{(1,1)}\) is the \((2, 4)\)-torus link, and \({\mathcal L}{(2,1)}\) is the (positive) Whitehead link. \({\mathcal L}_{(m,d)}\) is hyperbolic for \(m>1\) because it is the lifted link of a hyperbolic knot \({\mathcal L}_{(m,1)}\) to the \(d\)-fold unbranched covering space over the axis. (2)\({\mathcal L}_{(2,d)}\) is a chain link with axis of \((d+1)\) components. The \(d\)-component chain link is a sublink of \({\mathcal L}_{(2,d)}\) which does not include the axis.

The authors mainly deal with the case that \(m\) is even. Then \({\mathcal L}_{(m,d)}\) is a \((d+1)\)-component link. The first result of the article gives group presentations of both the exterior of \({\mathcal L}_{(m,d)}\) and the manifold obtained by surgery along the link with arbitrary coefficients. Since \({\mathcal L}_{(m,d)}\) is strongly invertible, the surgered manifold can be expressed as a \(2\)-fold branched covering space of \(S^3\) over a certain link. The method to obtain the branch link is well-known as “Montesinos trick”. The second result gives a branch link of the surgered manifold above as a \(2\)-fold branched covering space by the Montesinos trick.

For the results above, hyperbolicity is not needed. They also hold for the case that \({\mathcal L}_{(m,d)}\) and/or its surgered manifold are/is not hyperbolic. But, in order to say the “the results give a topological classification”, the isomorphism problem of the fundamental groups for the first result, or the uniqueness problem of the branched links for the second result should be solved. The authors should have mentioned this. However the article is interesting because the class \(\{{\mathcal L}_{(m,d)} \}\) includes many important links for Dehn surgery theory. To study the class is meaningful.

Let \(L_i\)\((i=1, \dots, d)\) be the \(1/m\)-rational tangle (i.e.\((-m)\)-half twists) with \(d\) and \(m\) positive integers. By arranging them linearly, we obtain a pretzel link consisting of \(d\) times \((-m)\)-half twists. Then there is an orientation-preserving autohomeomorphism of \(S^3\) with period \(d\) which fixes the pretzel link setwise, and maps \(L_i\) to \(L_{i+1}\) where \(L_{d+1}=L_1\). Then the link \({\mathcal L}_{(m,d)}\) is a union of the pretzel link and the axis of the periodic map. The link \({\mathcal L}_{(m,d)}\) is also periodic with period \(d\), and is strongly invertible along the axis perpendicular to all \(L_i\). We can also define \({\mathcal L}_{(m,d)}\) for \(m<0\) as the mirror image of \({\mathcal L}_{(-m,d)}\). The number of the components of \({\mathcal L}_{(m,d)}\) is \(2\) if both \(d\) and \(m\) are odd, \(3\) if \(d\) is even and \(m\) is odd, and \(d+1\) if \(m\) is even. There are two special families : (1)\({\mathcal L}_{(m,1)}\) is a \(2\)-bridge link of type \((2, m, -2)\) in Conway’s notation. \({\mathcal L}_{(1,1)}\) is the \((2, 4)\)-torus link, and \({\mathcal L}{(2,1)}\) is the (positive) Whitehead link. \({\mathcal L}_{(m,d)}\) is hyperbolic for \(m>1\) because it is the lifted link of a hyperbolic knot \({\mathcal L}_{(m,1)}\) to the \(d\)-fold unbranched covering space over the axis. (2)\({\mathcal L}_{(2,d)}\) is a chain link with axis of \((d+1)\) components. The \(d\)-component chain link is a sublink of \({\mathcal L}_{(2,d)}\) which does not include the axis.

The authors mainly deal with the case that \(m\) is even. Then \({\mathcal L}_{(m,d)}\) is a \((d+1)\)-component link. The first result of the article gives group presentations of both the exterior of \({\mathcal L}_{(m,d)}\) and the manifold obtained by surgery along the link with arbitrary coefficients. Since \({\mathcal L}_{(m,d)}\) is strongly invertible, the surgered manifold can be expressed as a \(2\)-fold branched covering space of \(S^3\) over a certain link. The method to obtain the branch link is well-known as “Montesinos trick”. The second result gives a branch link of the surgered manifold above as a \(2\)-fold branched covering space by the Montesinos trick.

For the results above, hyperbolicity is not needed. They also hold for the case that \({\mathcal L}_{(m,d)}\) and/or its surgered manifold are/is not hyperbolic. But, in order to say the “the results give a topological classification”, the isomorphism problem of the fundamental groups for the first result, or the uniqueness problem of the branched links for the second result should be solved. The authors should have mentioned this. However the article is interesting because the class \(\{{\mathcal L}_{(m,d)} \}\) includes many important links for Dehn surgery theory. To study the class is meaningful.

Reviewer: Teruhisa Kadokami (Shanghai)

### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

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

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\textit{S. H. Kim} and \textit{Y. Kim}, Int. J. Math. Math. Sci. 2010, Article ID 573403, 8 p. (2010; Zbl 1210.57018)

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