The colored Jones polynomial and Kontsevich–Zagier series for double twist knots

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots [Formula: see text] and [Formula: see text] where [Formula: see text] and [Formula: see text] are positive integers. In the [Formula: see text] case, this leads to new families of [Formula: see text]-hypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of [Formula: see text] gives a generalization of a duality at roots of unity between the Kontsevich–Zagier function and the generating function for strongly unimodal sequences.

Quantum modularity of partial theta series with periodic coefficients

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 F t ⁢ ( q ) \mathscr{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(3,2^{t}) , t ≥ 2 t\geq 2 , is a weight 3 2 \frac{3}{2} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ⁢ ( q ) F(q) .

Q-Series and quantum spin networks

The tail of a quantum spin network in the two-sphere is a [Formula: see text]-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by [Formula: see text]. We compute the [Formula: see text]-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure.

A diagrammatic approach to the AJ Conjecture

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $$\hat{A}$$ A ^ polynomial), with a classical invariant, namely the defining polynomial A of the $${\mathrm {PSL}_2(\mathbb {C})}$$ PSL 2 ( C ) character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors of the $$\hat{A}$$ A ^ -polynomial (after we set $$q=1$$ q = 1 , and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the $$\hat{A}$$ A ^ -polynomial that depends on a planar diagram of a knot (that conjecturally agrees with the $$\hat{A}$$ A ^ -polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate.

The slope conjecture for Montesinos knots

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

On the 2-head of the colored Jones polynomial for pretzel knots

In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.

Stability properties of the colored Jones polynomial

It is known that the colored Jones polynomial of a [Formula: see text]-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the [Formula: see text]-adequate link complement. We show that a power series similar to the tail of the colored Jones polynomial for [Formula: see text]-adequate links can be defined for all links, and that it is trivial if and only if the link is non [Formula: see text]-adequate.

A result on the Slope conjectures for 3-string Montesinos knots

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.

Normal and Jones surfaces of knots

We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open questions. We establish a relation between the Jones period of a knot and the number of sheets of the surfaces that satisfy the Strong Slope Conjecture (Jones surfaces). We also present numerical and experimental evidence supporting a stronger such relation which we state as an open question.