Essential tori in 3-manifolds not detected in any characteristic

TL;DR

This paper constructs examples of 3-manifolds with essential tori that cannot be detected by algebraic methods involving character varieties over any algebraically closed field.

Contribution

It introduces infinite families of 3-manifolds with essential tori undetectable by characteristic variety methods, highlighting limitations of current detection techniques.

Findings

Existence of infinite families of such 3-manifolds.

Essential tori not detectable via ideal points of $ ext{SL}_2( ext{F})$-character varieties.

Demonstrates limitations of algebraic detection methods in 3-manifold topology.

Abstract

Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters over any algebraically closed field $\mathbb{F}$.