Radial limit conjectures of quantum invariants

Quantum invariants are important topological objects motivated by physics. They are also important in number theory since they relates to modular forms. Such relations are formulated as “radial limit conjectures” in some cases. In this talk, I prove such conjectures for some cases with three key ideas: (1) To develop a new asymptotic formula by the Euler-Maclaurin summation formula; (2) To prove that the conjectures is deduced from the holomorphy of a certain rational function; (3) To prove the holomorphy by the induction on pruning of a plumbing graph.