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1. Volume Conjecture

This section has problems related to the Volume Conjecture

    1. The Colored Jones Polynomial Volume Conjecture for odd root of unity


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      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 n-th 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


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          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 n-th 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 Reshetikhin-Turaev and Turaev-Viro invariants for hyperbolic 3-manifolds


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              Problem 1.15.

              Let M be a hyperbolic 3-manifold. What are the asymptotics of the Turaev-Viro and Reshetikhin-Turaev invariants for this manifold?

                • Understanding connections between Baseilhac-Benedetti invariants and other quantum invariants


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                  Problem 1.2.

                  What are the connections between Baseilhac-Benedetti invariants and 1. ADO invariant? 2. BWY invariant? 3. modified TV invariant?
                    1. Remark. ADO invariant: Akutsu, Deguchi, Ohtsuki modified TV invariant: Geer-Patureau, 2010

                        • For links with diffeomorphic complements, how are their colored Jones polynomial (asymptotics) related?


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                          Problem 1.25.

                          If two links have diffeomorphic complements in $S^3$, relate the asymptotics of their colored Jones Polynomial.
                            1. Remark. Non-trivial example: whitehead link and twisted whitehead link.

                                • Pairs of 3-manifold having the same volume but different TV invariants


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                                  Problem 1.3.

                                  For two manifolds with same volume and different Turaev-Viro invariants, how are they related?

                                    • Teichmüller TQFT volume conjecture for Fundamental Shadow Link complements


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                                      Conjecture 1.35.

                                      Formulate and prove the volume conjecture coming from the Teichmüller TQFT for Fundamental Shadow Link complements
                                        1. Remark. Teichmuller TQFT reference: Anderson-Kashaev ICM, 2018, Anderson-Kashaev, arxiv 1305.4291 Fundamental Shadow Link complements reference: Costantino, D. Thurston, "3-manifold bound efficiently", J. Topology


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                                              Problem 1.4.

                                              Generalize Teichmüller TQFT to wider families e.g. cone manifolds or fundamental shadow link complements


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                                                  Problem 1.45.

                                                  Find a shadow formula for Teichmüller TQFT

                                                    • Two variable invariants and their connections


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                                                      Problem 1.5.

                                                      Are the Habiro 2-variable invariants for knots and Gukov-Manolescu invariant (also known as the F-invariant) the same?
                                                        1. Remark. Reference for Habiro invariant: "Unified WRT invariant for integral homology 3-spheres". Reference for F-invariant: "2 variable series for knot complements"


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                                                              Problem 1.55.

                                                              State and prove the volume conjecture for the Gukov-Manolescu invariant.

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