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We construct a geometric ideal triangulation for every fundamental shadowlink complement and solve the gluing equation explicitly in terms of theholonomies of the meridians of the link for any generic character in thedistinguished component of the $\mathrm{PSL}(2;\mathbb{C})$-character varietyof the link complement. As an application, we obtain a new formula for thevolume of a hyperideal tetrahedron in terms of its dihedral angles. Moreover,by using the ideal triangulation, we verify the 1-loop conjecture proposed byDimofte and Garoufalidis for every fundamental shadow link complement. We alsoprove a surgery formula for the 1-loop invariant with respect to certain niceideal triangulations of 3-manifolds with toroidal boundary.

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