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

Tushar Pandey, Ka Ho Wong

Published 2023-08-12Version 1

We construct a geometric ideal triangulation for every fundamental shadow link complement and solve the gluing equation explicitly in terms of the 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 an application, we obtain a new formula for the volume of a hyperideal tetrahedron in terms of its dihedral angles. Moreover, by using the ideal triangulation, we verify the 1-loop conjecture proposed by Dimofte and Garoufalidis for every fundamental shadow link complement. We also prove a surgery formula for the 1-loop invariant with respect to certain nice ideal triangulations of 3-manifolds with toroidal boundary.