Regular genus conjecture for n-dimensional tori

Prove that the regular genus of the n-dimensional torus $(\mathbb{S}^1)^n$ equals $1+\frac{(n+1)!(n-3)}{8}$ for every integer $n\ge 5$, and establish that the explicitly constructed $(n+1)$-regular colored graph with $(n+1)!$ vertices is a genus-minimal crystallization.

Background

The authors construct an (n+1)(n+1)-regular colored graph with (n+1)!(n+1)! vertices and show that its regular genus has the stated value for n≥4.Theyverifythatthegraphrepresentsthen−dimensionaltorusfordimensionsuptofour,butleavethegenus−minimalityandthecorrespondingtorusrealizationunresolvedforalln\ge 4. They verify that the graph represents the n-dimensional torus for dimensions up to four, but leave the genus-minimality and the corresponding torus realization unresolved for alln\ge 5$.

References

This lead us to the following conjecture.

— Regular genus of $\mathbb{S}^2 \times \mathbb{S}^1 \times \mathbb{S}^1$, $4$-torus, and small covers over $Δ^2 \times Δ^2$  (2506.01315 - Agarwal et al., 2 Jun 2025) in Conjecture 1, following the discussion of the n-dimensional torus