A knot
in
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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