# Modular Forms and Quantum Knot Invariants (18w5007)

Arriving in Banff, Alberta Sunday, March 11 and departing Friday March 16, 2018

## Organizers

Jeremy Lovejoy (CNRS, Université Paris 7)

Kazuhiro Hikami (Kyushu University)

Robert Osburn (University College Dublin)

## Objectives

The goal of this intense five-day workshop is to bring together international experts and young researchers in low-dimensional topology, number theory, string theory, quantum physics, algebraic geometry, conformal field theory, special functions and automorphic forms to discuss new developments and investigate potential directions for future research at the crossroads of modular forms and quantum knot invariants. A brief description of some topics of interest are as follows.

(1) The Quantum Modularity and Volume conjectures:

Quantum invariants of knots and 3-manifolds have been constructed rather combinatorially based on quantum groups,and their geometrical meanings are still unclear. The key is the Volume Conjecture, proposed by Kashaev and Murakami-Murakami. The Volume Conjecture was proved for some knots such as the figure-eight knot and torus knots, but is still open for arbitrary $K$.

The relationship between quantum invariants and the hyperbolic volume motivated Zagier to propose the notion of a "quantum modular form'" as a function having nice properties at a root of unity. A typical example is the Kontsevich-Zagier series; it coincides, at a root of unity, with the Kashaev invariant for the trefoil, and furthermore can be written in terms of the Eichler integral of the Dedekind eta-function. One of our goals is to analyze the asymptotic behavior of the Kashaev invariants for hyperbolic knots and to clarify their quantum modularity. These are important problems from the viewpoint of quantum topology and modular forms.

(2) Modularity of WRT invariants:

A unified WRT invariant for integral homology spheres was recently proposed by Habiro. Interestingly, the computation of this unified WRT invariant for certain manifolds leads to new mock theta functions. For example, a result of Bringmann, Hikami and Lovejoy says that a certain $q$-series $\phi(q)$ is a mock theta function and $\phi(-q^{1/2})$ is (up to an explicit factor) the unified WRT invariant of the Seifert manifold $\Sigma(2, 3, 8)$ which arises from $+2$ surgery on the trefoil knot.

One goal would be to study the unified WRT invariants of manifolds which arise from rational surgeries on other kinds of knots. These are typically quantum modular forms when viewed as functions at roots of unity, but when they converge inside the unit circle their behavior can vary. In some cases one obtains the usual indefinite binary theta functions related to mock theta functions, but in other cases one has less familiar expressions involving indefinite ternary quadratic forms or "partial'' positive definite forms. An important task is to make the modularity properties of these unified WRT invariants explicit.

(3) Stability:

In 2006, Dasbach and Lin observed stability in the coefficients of the colored Jones polynomial for an alternating knot $K$. This observation and its consequences have sparked a flurry of activity in both number theory and quantum topology. A few highlights are a resolution of this conjecture due to Armond and (independently) L$\hat{e}$ and Garoufalidis, the study of Rogers-Ramanujan type identities for "tails" of the colored Jones polynomial from the perspectives of skein theory and $q$-series and the study of potential stability of coefficients for generalizations of the colored Jones polynomial, namely for colored HOMFLY polynomials and colored superpolynomials.

(4) Explicit methods for $q$-hypergeometric series:

Quantum knot invariants are often expressible in terms of $q$-hypergeometric series, and explicit methods for these series are crucial. To give just two examples, the recent computation of the cyclotomic expansion of the colored Jones polynomial for the torus knots ($2,2t+1$) depended on key advances in the method of Bailey pairs, while the proofs of conjectures of Garoufalidis, L$\hat{e}$ and Zagier on the tails of alternating knots relied heavily on the Andrews-Bowman generalization of Sears' identity. Further development of these techniques is still needed, especially with a view toward applications to properties of knot invariants.

(1) The Quantum Modularity and Volume conjectures:

Quantum invariants of knots and 3-manifolds have been constructed rather combinatorially based on quantum groups,and their geometrical meanings are still unclear. The key is the Volume Conjecture, proposed by Kashaev and Murakami-Murakami. The Volume Conjecture was proved for some knots such as the figure-eight knot and torus knots, but is still open for arbitrary $K$.

The relationship between quantum invariants and the hyperbolic volume motivated Zagier to propose the notion of a "quantum modular form'" as a function having nice properties at a root of unity. A typical example is the Kontsevich-Zagier series; it coincides, at a root of unity, with the Kashaev invariant for the trefoil, and furthermore can be written in terms of the Eichler integral of the Dedekind eta-function. One of our goals is to analyze the asymptotic behavior of the Kashaev invariants for hyperbolic knots and to clarify their quantum modularity. These are important problems from the viewpoint of quantum topology and modular forms.

(2) Modularity of WRT invariants:

A unified WRT invariant for integral homology spheres was recently proposed by Habiro. Interestingly, the computation of this unified WRT invariant for certain manifolds leads to new mock theta functions. For example, a result of Bringmann, Hikami and Lovejoy says that a certain $q$-series $\phi(q)$ is a mock theta function and $\phi(-q^{1/2})$ is (up to an explicit factor) the unified WRT invariant of the Seifert manifold $\Sigma(2, 3, 8)$ which arises from $+2$ surgery on the trefoil knot.

One goal would be to study the unified WRT invariants of manifolds which arise from rational surgeries on other kinds of knots. These are typically quantum modular forms when viewed as functions at roots of unity, but when they converge inside the unit circle their behavior can vary. In some cases one obtains the usual indefinite binary theta functions related to mock theta functions, but in other cases one has less familiar expressions involving indefinite ternary quadratic forms or "partial'' positive definite forms. An important task is to make the modularity properties of these unified WRT invariants explicit.

(3) Stability:

In 2006, Dasbach and Lin observed stability in the coefficients of the colored Jones polynomial for an alternating knot $K$. This observation and its consequences have sparked a flurry of activity in both number theory and quantum topology. A few highlights are a resolution of this conjecture due to Armond and (independently) L$\hat{e}$ and Garoufalidis, the study of Rogers-Ramanujan type identities for "tails" of the colored Jones polynomial from the perspectives of skein theory and $q$-series and the study of potential stability of coefficients for generalizations of the colored Jones polynomial, namely for colored HOMFLY polynomials and colored superpolynomials.

(4) Explicit methods for $q$-hypergeometric series:

Quantum knot invariants are often expressible in terms of $q$-hypergeometric series, and explicit methods for these series are crucial. To give just two examples, the recent computation of the cyclotomic expansion of the colored Jones polynomial for the torus knots ($2,2t+1$) depended on key advances in the method of Bailey pairs, while the proofs of conjectures of Garoufalidis, L$\hat{e}$ and Zagier on the tails of alternating knots relied heavily on the Andrews-Bowman generalization of Sears' identity. Further development of these techniques is still needed, especially with a view toward applications to properties of knot invariants.