Asymptotic expansion of relative quantum invariants

Abstract

We propose Asymptotic Expansion Conjectures of the relative Reshetikhin-Turaev invariants, of the relative Turaev-Viro invariants and of the discrete Fourier transforms of the quantum 6j-symbols, and prove them for families of special cases. The significance of these expansions is that we do not specify the way that the sequence of the colorings converges to the limit. As a consequence, the terms in the expansion will have to depend on the index r, but the dependence is in a way that the terms are purely geometric invariants of the metrics on the underlying manifold and only the metrics vary with r.

On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs

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We verify the leading order term in the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants proposed in [56] for all pairs (M, L) satisfying the properties that M L is homeomorphic to some fundamental shadow link complement and the 3-manifold M is obtained by doing rational Dehn filling on some boundary components of the fundamental shadow link complement, under the assumptions that the denominator of the surgery coefficients are odd and the cone angles are sufficiently small. In particular, the asymptotics of the invariants captures the complex volume and the twisted Reidemeister torsion of the manifold M L associated with the hyperbolic cone structure determined by the sequence of colorings of the framed link L.

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“…In Section 2, we give a brief review for the preliminary knowledge required for the proof of Theorem 1.3. The materials in this section can be found in [53,55,56]. In Section 3, we compute the relative Reshetikhin-Turaev invariants of (M, L) and express it as a sum of the evaluation of certain holomorphic function at some integral points (Proposition 3.3).…”

“…Furthermore, in [56], T. Yang and the second author refined Conjectue 1.1 by studing the asymptotic expansion formula of the relative Reshetikhin-Turaev invariants. Let M be a closed oriented 3-manifold and let L be a framed hyperbolic link in M with n components.…”

“…In [56], Conjecture 1.2 has been proved for all pairs (M, L) obtained by doing a change-of-pair operation from the pair (M c , L FSL ) with sufficiently small cone angles. Similar to the relationship between Conjecture 1.1 and the Chen-Yang's volume conjecture, Conjecture 1.2 also provides a new approach to understand the asymptotic expansion of the Reshetikhin-Turaev invariants for closed oriented 3-manifold discussed in [16] and [36].…”

“…In this paper, we combine the ideas in [53], [54] and [56] to study the asymptotic expansion formula for the relative Reshetikhin-Turaev invariants of the pair (M, L) with M L homeomorphic to some (M c , L FSL ). Precisely, we let n ∈ N, I ⊆ {1, 2, .…”