Guo, Xiao; Zhang, Yu Virtually fibred Montesinos links of type \(\tilde {SL_2}\). (English) Zbl 1175.57006 Topology Appl. 156, No. 8, 1510-1533 (2009). Theorem: If a classic Montesinos link \(K\) has a cyclic rational tangle decomposition of the form \((q_1/p,\dots \, q_n/p)\) with \(p \geq 3\) odd, then \(K\) is virtually fibred. Here, virtually fibred means that the exterior has a finite cover which is a surface bundle over the circle. A link in the \(3\)-sphere is a classic Montesinos link if its double branched cover is a Seifert fibred manifold (total space of a circle bundle) such that no component of the branched set is a fibre. The proof of the above theorem relies on and generalizes the methods of I. Agol, S. Boyer and X. Zhang [J. Topol. 1, No. 4, 993–1018 (2008; Zbl 1168.57004)] for the special case that \(n\) is a multiple of \(p\). These results hopefully pave the way towards a proof of Thurston’s famous virtually fibred conjecture. Reviewer: Dieter Erle (Dortmund) Cited in 1 ReviewCited in 1 Document MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 57M12 Low-dimensional topology of special (e.g., branched) coverings Keywords:Montesinos link; \(3\)-manifold; \(\widetilde{SL}_2\)-geometry; classic Montesinos link; tangle; Seifert fibration; surface bundle; branched covering Citations:Zbl 1168.57004 PDFBibTeX XMLCite \textit{X. Guo} and \textit{Y. Zhang}, Topology Appl. 156, No. 8, 1510--1533 (2009; Zbl 1175.57006) Full Text: DOI arXiv References: [1] Ian Agol, Steven Boyer, Xingru Zhang, Virtually fibred Montesinos links, J. Topology, in press; Ian Agol, Steven Boyer, Xingru Zhang, Virtually fibred Montesinos links, J. Topology, in press · Zbl 1168.57004 [2] Walsh, G., Great circle links and virtually fibred knots, Topology, 44, 5, 947-958 (2005) · Zbl 1087.57005 [3] Wang, S.; Yu, F., Graph manifolds with non-empty boundary are covered by surface bundles, Math. Proc. Cambridge Philos. Soc., 122, 447-455 (1997) · Zbl 0899.57011 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.