  We seek to connect ideas in the theory of bridge trisections with other
well-studied facets of classical knotted surface theory. First, we show how the
normal Euler number can be computed from a tri-plane diagram, and we use this
to give a trisection-theoretic proof of the Whitney-Massey Theorem, which
bounds the possible values of this number in terms of the Euler characteristic.
Second, we describe in detail how to compute the fundamental group and related
invariants from a tri-plane diagram, and we use this, together with an analysis
of bridge trisections of ribbon surfaces, to produce an infinite family of
knotted spheres that admit non-isotopic bridge trisections of minimal
complexity.
