  This article re-examines a classic question in liquid-crystal physics: What
are the elastic modes of a nematic liquid crystal? The analysis uses a recent
mathematical construction, which breaks the director gradient tensor into four
distinct types of mathematical objects, representing splay, twist, bend, and a
fourth deformation mode. With this construction, the Oseen-Frank free energy
can be written as the sum of squares of the four modes, and saddle-splay can be
regarded as bulk rather than surface elasticity. This interpretation leads to
an alternative way to think about several previous results in liquid-crystal
physics, including: (1) free energy balance between cholesteric and blue
phases, (2) director deformations in hybrid-aligned-nematic cells, (3)
spontaneous twist of achiral liquid crystals confined in a torus or cylinder,
and (4) curvature of smectic layers.
