Left-orderability for surgeries on twisted torus knots. (English) Zbl 1417.57016

The L-space conjecture claims that an irreducible rational homology \(3\)-sphere is an L-space if and only if its fundamental group is not left-orderable. A knot is called an L-space knot if it admits a positive Dehn surgery yielding an L-space. It is well known that \(p/q\)-surgery on an L-space knot \(K\) yields an L-space if and only if \(p/q\ge 2g(K)-1\), where \(g(K)\) is the genus of \(K\).
For example, K. L. Baker and A. H. Moore [J. Math. Soc. Japan 70, No. 1, 95–110 (2018; Zbl 1390.57002)] showed that the \((-2,3,2n+1)\)-pretzel knots with \(n\ge 3\) and their mirror images are the only hyperbolic L-space knots among Montesinos knots. Recently, Z. Nie [Topology Appl. 261, 1–6 (2019; Zbl 1421.57011)] showed that \(p/q\)-surgery on the \((-2,3,2n+1)\)-pretzel knot with \(n\ge 3\) yields a \(3\)-manifold whose fundamental group is not left-orderable if \(p/q\ge 2n+3\), and left-orderable if \(p/q\) is sufficiently close to \(0\). We remark that its genus is \(n+2\).
The purpose of the paper under review is to extend the above result by Nie to a larger family of L-space knots. Let \(K\) be the \((n-2)\)-twisted \((3,3m+2)\)-torus knot, where \(m,n\ge 1\). More precisely, \(K\) is obtained from the \((3,3m+2)\)-torus knot by adding \((n-2)\)-full twists on an adjacent pair of strings. Some authors use the notation \(K(3,3m+2;2,n-2)\). We remark that \(g(K)=n+3m-1\) and that if \(m=1\) then \(K\) is the \((-2,3,2n+1)\)-pretzel knot.
The main theorem states that \(p/q\)-surgery on \(K\) yields a \(3\)-manifold whose fundamental group is not left-orderable if \(p/q\ge 2g(K)-1\), and left-orderable if \(p/q\) is sufficiently close to \(0\). For the first part, the argument is the same as the one in [loc. cit.]. Using the presentation of the fundamental group \(G\) of the resulting manifold by Dehn surgery, the assumption that \(G\) is left-orderable implies the existence of a monomorphism from \(G\) to \(\mathrm{Homeo}^+(\mathbb{R})\) without global fixed point. This leads to a contradiction. For the second part, the author shows that the Alexander polynomial of \(K\) has a simple root on the unit circle by a calculation. Then the conclusion follows from the result by M. Culler and N. M. Dunfield [Geom. Topol. 22, No. 3, 1405–1457 (2018; Zbl 1392.57012)] and C. Herald and X. Zhang [Proc. Am. Math. Soc. 147, No. 7, 2815–2819 (2019; Zbl 1516.57010)].


57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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