Characterizing slopes for satellite knots.(English)Zbl 1548.57018
For a given knot \(K\) in the \(3\)-sphere \(S^3\), we say that \(p/q\) is 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 \(K\).
For the unknot, the trefoil knots and the figure eight knot, all non-trivial slopes are known to be characterizing slopes; see [P. B. Kronheimer et al., Ann. Math. (2) 165, No. 2, 457–546 (2007;Zbl 1204.57038) andP. Ozsváth andZ. Szabó, J. Symplectic Geom. 17, No. 1, 251–265 (2019;Zbl 1444.57007)].M. Lackenby [Math. Ann. 374, No. 1–2, 429–446 (2019;Zbl 1421.57009)] showed that for any hyperbolic knot, slopes \(p/q\) are characterizing slopes provided \(|q|\) is sufficiently large. For torus knots,D. McCoy [Commun. Anal. Geom. 28, No. 7, 1647–1682 (2020;Zbl 1468.57006)] showed that there are only finitely many non-integral, non-characterizing slopes.
In the paper under review, the author establishes that any slope \(p/q\) is characterizing for any satellite knot provided \(|q|\) is sufficiently large. Hence, together with the results of Lackenby and McCoy, any slope \(p/q\) is characterizing for any knot provided \(|q|\) is sufficiently large. Furthermore, for composite knots \(K\), the author proves that every non-integral slope is characterizing for \(K\). As a corollary to this result, for infinitely many composite knots given in [K. L. Baker andK. Motegi, Algebr. Geom. Topol. 18, No. 3, 1461–1480 (2018;Zbl 1422.57010)] the author proves that the set of non-characterizing slopes consists of all integral slopes. It should be mentioned that these are the first examples of knots for which the complete list of non-characterizing slopes is determined and is not empty.
For the unknot, the trefoil knots and the figure eight knot, all non-trivial slopes are known to be characterizing slopes; see [P. B. Kronheimer et al., Ann. Math. (2) 165, No. 2, 457–546 (2007;Zbl 1204.57038) andP. Ozsváth andZ. Szabó, J. Symplectic Geom. 17, No. 1, 251–265 (2019;Zbl 1444.57007)].M. Lackenby [Math. Ann. 374, No. 1–2, 429–446 (2019;Zbl 1421.57009)] showed that for any hyperbolic knot, slopes \(p/q\) are characterizing slopes provided \(|q|\) is sufficiently large. For torus knots,D. McCoy [Commun. Anal. Geom. 28, No. 7, 1647–1682 (2020;Zbl 1468.57006)] showed that there are only finitely many non-integral, non-characterizing slopes.
In the paper under review, the author establishes that any slope \(p/q\) is characterizing for any satellite knot provided \(|q|\) is sufficiently large. Hence, together with the results of Lackenby and McCoy, any slope \(p/q\) is characterizing for any knot provided \(|q|\) is sufficiently large. Furthermore, for composite knots \(K\), the author proves that every non-integral slope is characterizing for \(K\). As a corollary to this result, for infinitely many composite knots given in [K. L. Baker andK. Motegi, Algebr. Geom. Topol. 18, No. 3, 1461–1480 (2018;Zbl 1422.57010)] the author proves that the set of non-characterizing slopes consists of all integral slopes. It should be mentioned that these are the first examples of knots for which the complete list of non-characterizing slopes is determined and is not empty.
Reviewer: Kimihiko Motegi (Tōkyō)
Cite
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