On the Volume Conjecture for hyperbolic Dehn-filled 3-manifolds along the twist knots

For a twist knot ${\mathcal{K}}_{p^{\mathrm{\prime }}}$, let $M$ be the closed $3$-manifold obtained by doing $(p,q)$ Dehn-filling along ${\mathcal{K}}_{p^{\mathrm{\prime }}}$. In this article, we prove that Chen-Yang’s volume conjecture holds for sufficiently large $\mathrm{\mid }p\mathrm{\mid }+\mathrm{\mid }q\mathrm{\mid }$ and $\mathrm{\mid }{p}^{\mathrm{\prime }}\mathrm{\mid }$ for $M$. In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston’s triangulation. Our triangulation has led to some new discoveries regarding symmetry, including insights into ``sister manifolds’’ as introduced by Hodgson, Meyerhoff, and Weeks.