Minimal contact triangulations of 3-manifolds

TL;DR

This paper investigates the minimal number of vertices needed for contact triangulations on 3-manifolds, providing explicit examples, bounds related to the $d^3$ invariant, and detailed analysis for specific overtwisted structures.

Contribution

It establishes a linear upper bound on the vertices of minimal contact triangulations in terms of the $d^3$ invariant and offers explicit examples and detailed analysis for certain overtwisted structures.

Findings

Vertex count grows at most linearly with the $d^3$ invariant.

Provided explicit examples of near-minimal contact triangulations.

Detailed study of contact triangulations on 3-torus.

Abstract

In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted contact structures grows at most linearly with respect to the relative $d^3$ invariant. We conjecture that this bound is optimal. We also discuss, in great details, contact triangulations for a certain family of overtwisted contact structures on 3-torus.