A parity constraint on the topological numbers of self-complementary triply-periodic surfaces

The stability zone of Novikov’s problem of planar sections of triply periodic surfaces are labeled by indivisible triples of integers, which we call the soul of the zone. Which triples may occur is not known. We show that, if a Morse function $f$ on ${\mathbb{T}}^3$ admits a translation $\tau (x)=x+v$ with $f\circ \tau =-f$, so that the sets $\{f<0\}$ and $\{f>0\}$ are congruent, then in the generic genus-$3$ situation every topological number $L$ satisfies $⟨L,2v⟩\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}2)$. The mechanism is that the electron/hole type of a null-homologous closed trajectory is invariant under the period lattice and is reversed by $\tau$. For $f=\cos 2\pi x+\cos 2\pi y+\cos 2\pi z$ and the $\mu$-cube, this means that the sum of the components of each soul is odd.