Geometry of fundamental shadow link complements and applications to the 1-loop conjecture

TL;DR

This paper constructs explicit geometric triangulations for fundamental shadow link complements, derives volume formulas, and verifies the 1-loop conjecture for a broad class of hyperbolic 3-manifolds using these triangulations.

Contribution

It provides explicit ideal triangulations and solves gluing equations for fundamental shadow link complements, enabling verification of the 1-loop conjecture in new cases.

Findings

Derived a new volume formula for hyperideal tetrahedra.

Verified the 1-loop conjecture for all fundamental shadow link complements.

Proved the invariance and surgery formula for the 1-loop invariant.

Abstract

We construct a geometric ideal triangulation for every fundamental shadow link complement and solve the gluing equation explicitly in terms of the logarithmic holonomies of the meridians of the link for any generic character in the distinguished component of the $\mathrm{PSL}(2;\mathbb{C})$-character variety of the link complement. As immediate applications, we obtain a new formula for the volume of a hyperideal tetrahedron in terms of its dihedral angles, and a formula for the volume of hyperbolic 3-manifolds obtained by doing Dehn-fillings to some of the boundary components of fundamental shadow link complements. Moreover, by using these ideal triangulations, we verify the 1-loop conjecture proposed by Dimofte and Garoufalidis for every fundamental shadow link complement. By using the result of Kalelkar-Schleimer-Segerman \cite{KSS}, we also prove the topological invariance of the 1-loop invariant and show that the 1-loop invariant satisfies a surgery formula. As a result, we prove the 1-loop conjecture for manifolds obtained by doing sufficiently long Dehn-fillings on boundary components of any fundamental shadow link complement. This verifies the 1-loop conjecture for a large class of hyperbolic 3-manifolds.