[2103.04254] Adjoint Torsion Leidemeister Torsion and Gram Matrices

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overview: Computes the adjoint torsion Leidemeister torsion of closed oriented hyperbolic $3$ manifolds and hyperbolic $3$ manifolds with toroidal boundaries. Our formula considers the manifold obtained by Dane filling along the appropriate boundary component of the underlying shadow-link complement, and the formula is based on the meridian logarithmic holonomy of the boundary component. As an important special case, we also write down the formula for the adjoint torsion Leidemeister torsion for twice the hyperbolic $3$ manifold with perfect geodesic boundaries in terms of the edge lengths of the geometric ideal triangulation of the manifold. These unexpected formulas were inspired by the study of the asymptotic expansion of quantum invariants and played an important role\,\cite{WY}.