Let
be a bundle
over
with fiber
a
–manifold
and with monodromy
. Gay and Kirby
showed that if
fixes
a genus
Heegaard
splitting of
then
has a genus
trisection.
Genus
trisections have been found in certain special cases, such as the case where
is trivial, and it is known that trisections of genus lower than
cannot
exist in general. We generalize these results to prove that there exists a trisection of genus
whenever
fixes a genus
Heegaard
surface of
. This
means that
can be nontrivial, and can preserve or switch the two handlebodies
of the Heegaard splitting. We additionally describe an algorithm to
draw a diagram for such a trisection given a Heegaard diagram for
and a
description of
.
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