1. Volume Conjecture
This section has problems related to the Volume Conjecture
The Colored Jones Polynomial Volume Conjecture for odd root of unity
Conjecture 1.05.
Let $L \in S^3$ be a hyperbolic link. For n odd, let $J_n(L) \in \mathbb{Z}[t^{\pm}]$ be the nth normalized Colored Jones Polynomial. Then, as $n \rightarrow \infty$, $$ J_n(L, t = e^{\frac{2\pi i}{n}}) \sim \text{exp} \left[\frac{n}{2\pi} (\text{vol}(S^3  L) + i CS(S^3  L) )\right] $$ 
The Colored Jones Polynomial Volume Conjecture for even root of unity
Conjecture 1.1.
Let $L \in S^3$ be a hyperbolic link. For n even, let $J_n(L) \in \mathbb{Z}[t^{\pm}]$ be the nth normalized Colored Jones Polynomial. Then, as $n \rightarrow \infty$, $$ J_n(L, t = e^{\frac{2\pi i}{n}}) \sim \text{exp} \left[\frac{n}{2\pi} (\text{vol}(S^3  L) + i CS(S^3  L) )\right] $$ 
Asymptotics of ReshetikhinTuraev and TuraevViro invariants for hyperbolic 3manifolds
Problem 1.15.
Let M be a hyperbolic 3manifold. What are the asymptotics of the TuraevViro and ReshetikhinTuraev invariants for this manifold? 
Understanding connections between BaseilhacBenedetti invariants and other quantum invariants
Problem 1.2.
What are the connections between BaseilhacBenedetti invariants and 1. ADO invariant? 2. BWY invariant? 3. modified TV invariant?
Remark. ADO invariant: Akutsu, Deguchi, Ohtsuki modified TV invariant: GeerPatureau, 2010


For links with diffeomorphic complements, how are their colored Jones polynomial (asymptotics) related?
Problem 1.25.
If two links have diffeomorphic complements in $S^3$, relate the asymptotics of their colored Jones Polynomial.
Remark. Nontrivial example: whitehead link and twisted whitehead link.


Pairs of 3manifold having the same volume but different TV invariants
Problem 1.3.
For two manifolds with same volume and different TuraevViro invariants, how are they related? 
Teichmüller TQFT volume conjecture for Fundamental Shadow Link complements
Conjecture 1.35.
Formulate and prove the volume conjecture coming from the Teichmüller TQFT for Fundamental Shadow Link complements
Remark. Teichmuller TQFT reference: AndersonKashaev ICM, 2018, AndersonKashaev, arxiv 1305.4291 Fundamental Shadow Link complements reference: Costantino, D. Thurston, "3manifold bound efficiently", J. Topology


Problem 1.4.
Generalize Teichmüller TQFT to wider families e.g. cone manifolds or fundamental shadow link complements 
Two variable invariants and their connections
Problem 1.5.
Are the Habiro 2variable invariants for knots and GukovManolescu invariant (also known as the Finvariant) the same?
Remark. Reference for Habiro invariant: "Unified WRT invariant for integral homology 3spheres". Reference for Finvariant: "2 variable series for knot complements"

Cite this as: AimPL: Quantum invariants and lowdimensional topology, available at http://aimpl.org/quantumlowdimtop.