Reconstructing surface triangulations by their intersection matrices

TL;DR

This paper proves that the intersection matrix of a finite simplicial complex uniquely determines the triangulation of a closed connected surface, up to isomorphism.

Contribution

It establishes that the intersection matrix fully encodes the surface triangulation, providing a new way to reconstruct surfaces from combinatorial data.

Findings

Intersection matrix determines surface triangulation up to isomorphism

Provides a method for reconstructing surfaces from intersection data

Enhances understanding of combinatorial surface invariants

Abstract

The intersection matrix of a finite simplicial complex has as each of its entries the rank of the intersection of its respective simplices. We prove that such matrix defines the triangulation of a closed connected surface up to isomorphism.