Quantum modularity of partial theta series with periodic coefficients

Abstract

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series ${\mathcal{F}}_t(q)$ which matches (at a root of unity) the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t\ge 2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series $F(q)$.

16 pages, to appear in Forum Mathematicum

References in corpus (8)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• 3d Modularity
• q-series and L-functions related to half-derivatives of the Andrews--Gordon identity
• Quantum Invariants, Modular Forms, and Lattice Points II
• APS $\eta$-invariant, path integrals, and mock modularity
• Three-Manifold Quantum Invariants and Mock Theta Functions
• Generalized Fishburn numbers and torus knots
• A note on additive twists, reciprocity laws and quantum modular forms

Cited by in corpus (3)

• Generalized Fishburn numbers and torus knots
• The colored Jones polynomial and Kontsevich-Zagier series for double twist knots
• Asymptotics and sign patterns for coefficients in expansions of Habiro elements