Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

A relative version of the Turaev–Viro invariants and the volume of hyperbolic polyhedral 3‐manifolds

“…The goal of this section is to prove Theorem 1.4. The main tool we use is Proposition 5.1, which is a generalization of the standard saddle point approximation [26] and whose proof can be found in [39,Appendix]. With a geometric preparation in Sections 5.1 and 5.…”

“…The goal of this section is to prove Theorem 1.4. The main tool we use is Proposition 5.1, which is a generalization of the standard saddle point approximation [26] and whose proof can be found in [39, Appendix]. With a geometric preparation in Sections 5.1 and 5.2, we in Section 5.3 estimate the leading Fourier coefficients, and, respectively, in Sections 5.4 and 5.5 estimate the nonleading Fourier coefficients and the error term in the formula of the relative Turaev–Viro invariants obtained in Propositions 4.2 and 4.3.…”

“…As an immediate consequence of Theorems 1.2 and 1.4, we have the following. Theorem The volume conjecture of the relative Reshetikhin–Turaev invariants [39, Conjecture 1.1] is true for all pairs $$(D(M),D(E))$$ with sufficiently small cone angles.…”

“…Theorem 1.5. The volume conjecture of the relative Reshetikhin-Turaev invariants [39,Conjecture 1.1] is true for all pairs (𝐷(𝑀), 𝐷(𝐸)) with sufficiently small cone angles.…”

“…Various quantum invariants are expected to contain geometric information of various threedimensional objects. See, for example, [4,8,17,24,39]. For the relative Turaev-Viro invariants, the corresponding geometric object is the hyperbolic polyhedral metric.…”