Special representations for \(n\)-bridge links (Q1330878)

The Schubert normal form of a 2-bridge link, \(K(p, q)\), consists of two overpasses (called `bridges') and two underpasses, and is characterized by two coprime integers \(p\), \(q\) \((1\leq p<q)\). Here \(p-1\) is the number of the undercrossings of each of the bridges, and \(q\) indicates the undercrossings to which the endpoints of the bridges are connected. Schubert classified all 2-bridge links in terms of this normal form. Generalizing this presentation, the authors show that any 3-bridge link can be represented by a special type of diagram that is characterized by six integers. They show that this presentation is not unique using some results of \textit{O. Morikawa} [Math. Semin. Notes, Kobe Univ. 9, 349-369 (1981; Zbl 0486.57005), Yokohama Math. J. 30, 53-72 (1982; Zbl 0526.57004)], who studied a class of 3-bridge links. Also, they have no general criterion for classifying 3-bridge links. It is known [\textit{J. S. Birman} and \textit{H. M. Hilden}, Trans. Am. Math. Soc. 213, 315-352 (1975; Zbl 0312.55004)] that any closed orientable 3- manifold with Heegaard genus two is a two-fold branched cover of \(S^3\) branched over a 3-bridge link. Using this presentation, the authors represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, they prove some results about these 3-manifolds and extend the 3-bridge presentation to \(n\)-bridge links.