Complementary legs and rational balls.

*(English)*Zbl 1429.57012One of the groups of 3-manifolds that has been intensively studied is the set of Seifert manifolds. They are orientable closed manifolds admitting a fixed point free action of \(S^1\) and are classified by their Seifert invariants.
All lens spaces are examples of Seifert manifolds.
Seifert manifolds admit an involution that presents them as double covers of \(S^3\) with branch set a link. This link can be easily recovered from any plumbing graph \(P\) that provides a surgery presentation for the Seifert manifold \(Y_P\).
The boundary of the surface obtained by plumbing bands according to \(P\) is called the Montesinos link.
A surface singularity is called ribbon if it is the identification of two segments, one contained in the interior of the surface and the other one joining two different boundary points.
One of the questions about a 3-manifold is whether or not it bounds a rational homology ball.
In the case of lens spaces, there is a complete classification, and there are partial answers to this question in case of Seifert manifolds.
Among the Seifert fibered rational homology spheres \(Y\) with Seifert invariants \(Y=Y(b;(\alpha_1,\beta_1),\dots,(\alpha_n,\beta_n))\), \(b\in\mathbb{Z}\), \(\alpha_i>\beta_i>0\), there are those having a pair of complementary legs, that is, a pair of invariants satisfying \(\frac{\beta_i}{\alpha_i}+\frac{\beta_j}{\alpha_j}=1\).
Topologically, it means that the boundaries of the manifolds obtained by plumbing along the linear graphs determined by the continued fraction expansions of \(\frac{\alpha_i}{\beta_i}\) and \(\frac{\alpha_j}{\beta_j}\) are two lens spaces that differ only in orientation.
For \(n\le 3\), there is a one-to-one correspondence between the set of Seifert manifolds and the set of the corresponding Montesinos links.

In this paper, the author studies the Seifert rational homology spheres with two complementary legs and presents a complete list \(\mathscr{L}\) of Seifert manifolds with three exceptional fibers and two complementary legs bounding rational homology balls. The author proves that the Seifert manifold \(Y=Y(b;(\alpha_1,\beta_1),(\alpha_2,\beta_2),(\alpha_3,\beta_3))\) with \(\frac{\beta_2}{\alpha_2}+\frac{\beta_3}{\alpha_3}=1\) is the boundary of a rational homology ball if and only if \(Y\) belongs to the list \(\mathscr{L}\). Also, the author shows that each Montesinos link in the set \(\mathscr{L}\) is the boundary of a ribbon surface \(\Sigma\) such that \(\chi(\Sigma)=1\).

In this paper, the author studies the Seifert rational homology spheres with two complementary legs and presents a complete list \(\mathscr{L}\) of Seifert manifolds with three exceptional fibers and two complementary legs bounding rational homology balls. The author proves that the Seifert manifold \(Y=Y(b;(\alpha_1,\beta_1),(\alpha_2,\beta_2),(\alpha_3,\beta_3))\) with \(\frac{\beta_2}{\alpha_2}+\frac{\beta_3}{\alpha_3}=1\) is the boundary of a rational homology ball if and only if \(Y\) belongs to the list \(\mathscr{L}\). Also, the author shows that each Montesinos link in the set \(\mathscr{L}\) is the boundary of a ribbon surface \(\Sigma\) such that \(\chi(\Sigma)=1\).

Reviewer: Andrew Bucki (Edmond)

##### MSC:

57K30 | General topology of 3-manifolds |

57K33 | Contact structures in 3 dimensions |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

##### Keywords:

Seifert manifold; rational homology sphere; Montesinos link; complementary legs; ribbon surface
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\textit{A. G. Lecuona}, Mich. Math. J. 68, No. 3, 637--649 (2019; Zbl 1429.57012)

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