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Virtually fibred Montesinos links of type \(\tilde {SL_2}\). (English) Zbl 1175.57006

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.

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

Citations:

Zbl 1168.57004
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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
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