We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3-manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3-manifolds with totally geodesic boundary.

中文翻译：

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Turaev–Viro 不变量的相对版本和双曲多面体 3-流形的体积

我们为具有非空边界和边缘着色的理想三角紧凑 3 流形定义了 Turaev-Viro 不变量的相对版本，概括了流形的 Turaev-Viro 不变量 [36]。我们还提出了这些不变量的体积猜想，这些不变量的渐近行为与双曲多面体度量中流形的体积有关[22, 23]，边缘的奇异轨迹和由着色确定的锥角，并证明了猜想锥角足够小的情况。这表明了解决由 Chen-Yang [8] 提出的具有完全测地线边界的双曲 3-流形的 Turaev-Viro 不变量的体积猜想的方法。