Volume 21, issue 6 (2021)

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Persistently foliar composite knots

Charles Delman and Rachel Roberts

Algebraic & Geometric Topology 21 (2021) 2761–2798

A knot κ in S3 is persistently foliar if, for each nontrivial boundary slope, there is a cooriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a cooriented taut foliation in every manifold obtained by nontrivial Dehn surgery on that knot. We show that any composite knot with a persistently foliar summand is persistently foliar and that any nontrivial connected sum of fibered knots is persistently foliar. As an application, it follows that any composite knot in which each of two summands is fibered or at least one summand is nontorus alternating or Montesinos is persistently foliar.

We note that, in constructing foliations in the complements of fibered summands, we build branched surfaces whose complementary regions agree with those of Gabai’s product disk decompositions, except for the one containing the boundary of the knot complement. It is this boundary region which provides for persistence.

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taut foliation, persistently foliar knot, L–space, L–space knot, composite knot, spine, branched surface
Mathematical Subject Classification 2010
Primary: 57M50
Received: 12 June 2019
Revised: 14 July 2020
Accepted: 12 August 2020
Published: 22 November 2021
Charles Delman
Department of Mathematics and Computer Science
Eastern Illinois University
Charleston, IL
United States
Rachel Roberts
Department of Mathematics and Statistics
Washington University in St Louis
St Louis, MO
United States