No 13-vertex triangulation of the three-torus
Year: 2026
Type: preprint
Abstract: Kühnel and Lassmann constructed a neighborly triangulation of the three-torus on 15 vertices, and Lutz conjectured that it is vertex-minimal. The standard lower-bound arguments exclude at most twelve vertices but do not decide the 13- and 14-vertex cases. We exclude the 13-vertex case. Our proof uses the Novik--Swartz $g_2$-inequality and a simplicial-complement reduction to a nine-vertex model of the punctured three-torus. By analyzing fundamental groups of small simplicial complexes and applying Alexander duality in a vertex link, we obtain a contradiction from the deficiency of $\Z^3$.
Source: arXiv (Cornell University)
Author: Lim, Geunho
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225830902
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