Tropical varieties associated to ideal triangulations: The Whitehead link complement

TL;DR

This paper presents a computational approach combining ideal triangulations, tropical geometry, and normal surface theory to analyze the Whitehead link complement, revealing new insights into its character variety and boundary detection.

Contribution

It introduces a novel computational method linking tropical geometry with 3-manifold topology, specifically applied to the Whitehead link complement.

Findings

Affine algebraic sets related to the Whitehead link complement are computed.

All boundary curves are strongly detected by the character variety.

The approach can be generalized to other cusped hyperbolic 3-manifolds.

Abstract

This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-normal surface theory is related to the space of incompressible surfaces and to the unit ball of the Thurston norm. It is shown that all boundary curves of the Whitehead link complement are strongly detected by its character variety. The specific results obtained can be used to study the geometry and topology of the Whitehead link complement and its Dehn surgeries. The methods can be applied to any cusped hyperbolic 3--manifold of finite volume.