Non-integer characterizing slopes for torus knots (Q2214598)

For a given knot \(K\) in the \(3\)-sphere \(S^3\), we call \(p/q\) a characterizing slope for \(K\) if whenever the result of \(p/q\)-surgery on a knot \(K'\) in \(S^3\) is orientation preservingly homeomorphic to the result of \(p/q\)-surgery on \(K\), then \(K'\) is isotopic to \(K\). \textit{P. B. Kronheimer} et al. [Ann. Math. (2) 165, No. 2, 457--546 (2007; Zbl 1204.57038)] proved that every nontrivial slope is a characterizing slope for the unknot, which settles the long-standing conjecture of Gordon. \textit{P. Ozsváth} and \textit{Z. Szabó} [J. Symplectic Geom. 17, No. 1, 251--265 (2019; Zbl 1444.57007)] extended this result to the trefoil and the figure-eight knot. \textit{Y. Ni} and \textit{X. Zhang} [Algebr. Geom. Topol. 14, No. 3, 1249--1274 (2014; Zbl 1297.57019)] studied characterizing slopes for torus knots, and proved that \(T_{5, 2}\) has only finitely many non-characterizing slopes which are not negative integers. In the paper under review, the author establishes a similar result for arbitrary torus knots, i.e. he proves that any torus knot \(T_{r, s}\) with \(r, s >1\) has only finitely many non-characterizing slopes which are not negative integers. In general, the author shows that if \(p/q\)-surgeries on \(K\) and \(K'\) result in the same oriented \(3\)-manifold for \(|p|\ge 12 + 4q^2 -2q + 4q g(K)\) and \(q\ge 3\), where \(g(K)\) denotes the genus of \(K\), then \(K\) and \(K'\) have the same Alexander polynomial and the same genus, and \(K\) is fibered if and only if \(K'\) is fibered. In another paper [Math. Res. Lett. 26, No. 5, 1517--1526 (2019; Zbl 1439.57023)], the author proves that any hyperbolic knot can have only finitely many non-characterizing slopes with \(q\ge 3\), strengthening a result of \textit{M. Lackenby} [Math. Ann. 374, No. 1--2, 429--446 (2019; Zbl 1421.57009)].