Unimodular Triangulations of 3D Lattice Polytopes

• Unimodular triangulations are decompositions of lattice polytopes into simplices of normalized volume one, underpinning resolutions in toric geometry and combinatorial optimization.
• The methodology employs empty tetrahedra classification, layered decompositions, and 'toblerone' prism techniques to achieve triangulations that are often standard on the boundary.
• Existence results confirm that for composite dilation factors k ≥ 6, unimodular triangulations exist, while the open cases at k = 3 and 5 highlight ongoing challenges.

A unimodular triangulation is a decomposition of a polytope or cone into simplices whose vertices are lattice points and whose normalized volume is one. In the context of dilated 3-dimensional lattice polytopes, determining the precise set of dilation factors $k$ for which the $k$-fold dilation $kP$ of a given polytope $P$ admits a unimodular triangulation is a fundamental question at the intersection of toric geometry, discrete geometry, and combinatorics. The existence of such triangulations has significant implications for the resolution of singularities in toric varieties, integer programming, and Ehrhart theory.

1. Core Results on Dilated 3-Polytopes

The principal result is that for every 3-dimensional lattice polytope $P$, the dilate $kP$ has a unimodular triangulation for all sufficiently large $k$. The work (Santos et al., 2013) sharpens this, demonstrating that for all $k \geq 6$, a unimodular triangulation of $kP$ exists, and that the set of such $k$ is both multiplicatively closed (i.e., contains all composite numbers) and forms an additive semigroup (if $k_1$ and $k_2$ are in the set, so is $k_1 + k_2$). As a consequence, for every 3-polytope, the dilation $kP$ can be triangulated into unimodular simplices for any composite $k$, and for $k$ equal to 7 or 11 via ad-hoc methods, although the standard boundary property is lost in those cases.

Specifically, the open cases are $k \in \{3, 5\}$, while $k=1$ and $k=2$ are proven not to work in general. For all other $k$, especially composites, unimodular triangulations exist.

2. Mathematical Techniques and Structural Framework

Classification of Empty Tetrahedra

The theory uses the classification of empty lattice tetrahedra (White’s classification) as building blocks. Every 3-dimensional empty lattice tetrahedron is unimodularly equivalent to

$$\Delta(p, q) = \operatorname{conv}\{ (0, 0, 0), (1, 0, 0), (0, 0, 1), (p, q, 1) \}$$

with $\gcd(p, q) = 1$. Through linear transformations, any empty tetrahedron can be brought to this form and, up to change of lattice, all such simplices have width 1 in some direction.

Boundary Compatibility and “Standard” Triangulations

A key technical notion is the property of being "standard on the boundary" (Definition 1.4). For a triangulation to be standard on the boundary, each facet of $k\Delta$ must be triangulated by its canonical unimodular triangulation, using only edges parallel to those of the original simplex. This ensures that triangulations of adjacent tetrahedra (as one glues together a triangulation of a whole polytope) remain compatible along shared faces. While results for $k = 7$ and $k = 11$ guarantee unimodular triangulations, these need not be standard on the boundary, which introduces technical complications for constructing global triangulations.

Layered Decompositions and “Toblerone” Prisms

The construction for general $k$ involves decomposing $k\Delta(p, q)$ into horizontal layers corresponding to fixed third coordinates. Each layer is further broken into "toblerone" (long, thin triangular prism) regions, which can be triangulated by monotone paths in a fundamental square of the lattice.

In more detail, certain maximal monotone lattice paths (maximal $X$-paths and $Y$-paths) in the fundamental square structure the triangulation. For a prism in the $(z = i)$ layer, one considers whether the $X$ and $Y$-paths are compatible (i.e., intersect properly). If so, one can guarantee that the maximal subprism is "tetragonal," admitting a particularly well-behaved unimodular triangulation.

Additive and Multiplicative Closure

The constructed triangulations are shown to be compatible under both addition and multiplication of the dilation factor $k$. That is, if triangulations for $k_1$ and $k_2$ (with standard boundary) exist, these can be combined to yield one for $k_1 + k_2$, and similarly for composite numbers via appropriate product decompositions. This closure property reduces the problem to a finite check on small prime $k$.

3. Implications and Applications

Toric Geometry and Resolution of Singularities

Unimodular triangulations correspond to crepant and projective desingularizations in toric geometry. For each 3-dimensional lattice polytope, a sufficiently large dilation ensures that its associated toric variety has a resolution of singularities via a regular fan supported by a unimodular triangulation (Santos et al., 2013).

Algorithmic and Combinatorial Consequences

Layered and prism-based decompositions provide explicit, constructive triangulations. These are foundational in discrete geometry, with relevance to integer programming, combinatorial optimization, and the computation of Ehrhart and $h^*$-polynomials, and for practical subdivision of polytopes into simplices of controlled combinatorial and geometric properties.

4. Challenges and Open Questions

Boundary Conditions and Nonstandard Triangulations

A persistent challenge is constructing unimodular triangulations that are standard on the boundary, required for gluing smaller triangulations to triangulate complex polytopes. For $k=7$ and $k=11$, standardness cannot always be ensured. For the open cases $k=3, 5$, there is no known construction, and their resolution likely demands fundamentally new ideas or a deeper understanding of obstructions at small scale.

Regularity

While the constructions always produce unimodular triangulations, in general they do not guarantee regularity (i.e., being induced by a convex piecewise-linear height function). Regularity is achieved for tetragonal tetrahedra (where $p = \pm1 \bmod q$), but not for all cases under these constructions.

5. Summary Table: Dilation Factor $k$ and Triangulation Existence

Dilation $k$	Unimodular Triangulation for all 3-Polytopes?	Standard on Boundary?	Known Issues
1	No	N/A	General counterexamples
2	No	N/A	General counterexamples
3	Open	N/A	No known construction
4	Yes	Yes
5	Open	N/A	No known construction
6, 8, ...	Yes	Yes	Composite $k$
7, 11	Yes	Not always	Nonstandard boundary needed

6. Context in Higher Dimensions and Future Directions

The findings raise the broader question of whether, for each dimension $n$, there exists a universal constant $k_n$ such that $k_n P$ admits a unimodular triangulation for any $n$-dimensional lattice polytope $P$. While the answer is affirmative in dimension three for $k \geq 6$ and all composite $k$, no finite universal $k_4$ is currently known in four dimensions, nor are such constants generally established in higher dimensions. The closure results for addition and multiplication in $k$ suggest analogous strategies may be fruitful, but the combinatorial and arithmetic complexities increase rapidly.

7. Conclusions

This work (Santos et al., 2013) provides a nearly complete characterization of dilation factors $k$ for which dilated 3-polytopes admit unimodular triangulations. The solution leverages the arithmetic and combinatorial structure of empty lattice tetrahedra, layer-based decompositions, and compatibility properties. Except for the unresolved small primes $k = 3$ and $k = 5$, the classification is definitive, and the techniques employed yield both existence results and explicit constructive methods with applications in several areas of mathematics and theoretical computer science.