New York Journal of Mathematics Volume 25 (2019), 518-540
 Volume 25 Volume Index

On L-space knots obtained from unknotting arcs in alternating diagrams

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 Published: June 20, 2019. Keywords: L-space, alternating diagram, unknotting crossing, branched double cover. Subject: 57M25, 57M27.

Abstract
Let D be a diagram of an alternating knot with unknotting number one. The branched double cover of S3 branched over D is an L-space obtained by half integral surgery on a knot KD. We denote the set of all such knots KD by $\mathcal D$. We characterize when KD ∈ $\mathcal D$ is a torus knot, a satellite knot or a hyperbolic knot. In a different direction, we show that for a given n > 0, there are only finitely many L-space knots in $\mathcal D$ with genus less than n.

Acknowledgements

We would like to thank Ken Baker, Josh Greene, Matt Hedden, John Luecke and Tom Mark for helpful conversations. We are also grateful to an anonymous referee for their detailed feedback.

Author information

Andrew Donald:
School of Mathematics
University of Bristol
Bristol, BS8 1TW, UK

andrew.donald@bristol.ac.uk

Duncan McCoy:
Department of Mathematics
University of Texas at Austin
Austin, TX 78712, USA

d.mccoy@math.utexas.edu

Faramarz Vafaee:
Department of Mathematics
Duke University
Durham, NC 27708, USA

vafaee@math.duke.edu