On left-orderability and cyclic branched coverings. (English) Zbl 1328.57017

Y. Hu [Algebr. Geom. Topol. 15, No. 1, 399–413 (2015; Zbl 1312.57001)] gives a sufficient condition for the fundamental group of the \(r\)-th cyclic branched cover of a knot to be left-orderable. Using this, she proves that for any two-bridge knot \(S(p,q)\) of Schubert’s notation with \(p\equiv 3\pmod{4}\), the fundamental group of the \(r\)-th cyclic branched cover is left-orderable for sufficiently large \(r\). The paper under review examines such a range of covering degree for double-twist knots, including genus one two-bridge knots. The simplest case is a double-twist knot \(J(2m,2n)\) with \(m,n>0\), which is the two-bridge knot \(S(4mn-1,2n)\). If the covering degree \(r\) is bigger than \(\pi/\arccos\sqrt{1-(4mn)^{-1}}\), then the \(r\)-fold cyclic branched cover of the knot has left-orderable fundamental group. Similar evaluations are also given for the other two classes \(J(2m+1,2n)\) and \(J(2m+1,-2n)\), with \(m,n>0\). The main idea is to find a non-abelian representation of the knot group in \(SL_2(\mathbb{R})\) which satisfies a certain condition, to take its lift to the universal covering group \(\widetilde{SL_2(\mathbb{R})}\), and to induce a non-trivial representation of the fundamental group of the cyclic branched cover in \(\widetilde{SL_2(\mathbb{R})}\), which is known to be left-orderable.


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)


Zbl 1312.57001
Full Text: DOI arXiv Euclid


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