5 problems

3-torus $f$-vector conjecture. The $f$-vectors of triangulations of the $3$-torus $T^3$ are characterized by

Kühn­el–Lassmann 3-torus conjecture. The Kühn­el–Lassmann triangulation of $T^3$ with $15$ vertices is vertex-minimal and is unique with

3-torus vertex-minimality conjecture. The $15$ vertices of the known $15$-vertex triangulation of $T^3$ are minimal for a triangulation of the $3$-dimensional torus.

Symmetric surface embedding conjecture. (1) All genera $g$ realizing the upper bound $E^+_+(\Sigma_g)=12(g-1)$ are listed in Theorem 1(1), and all examples realizing those genera a…

Let $T^3$ be the $3$-torus, and consider the specified neighborly $15$-vertex triangulation of $T^3$. 3-torus minimality conjecture. This triangulation is the unique vertex-minimal…