Euler characteristic and quadrilaterals of normal surfaces

TL;DR

This paper establishes inequalities connecting the Euler characteristic of normally embedded surfaces in 3-manifolds with their combinatorial normal quadrilaterals and triangles, linking topological invariants with combinatorial data.

Contribution

It introduces new inequalities that relate topological and combinatorial properties of normal surfaces in 3-manifolds, enhancing understanding of their structure.

Findings

Derived inequality relating Euler characteristic and normal quadrilaterals

Established bounds between normal triangles and quadrilaterals based on tetrahedron sharing

Provides a link between topological invariants and combinatorial descriptions

Abstract

Let $M$ be a compact 3-manifold with a triangulation $\tau$. We give an inequality relating the Euler characteristic of a surface $F$ normally embedded in $M$ with the number of normal quadrilaterals in $F$. This gives a relation between a topological invariant of the surface and a quantity derived from its combinatorial description. Secondly, we obtain an inequality relating the number of normal triangles and normal quadrilaterals of $F$, that depends on the maximum number of tetrahedrons that share a vertex in $\tau$.