Program

We show that the only way of changing the framing of a link by ambient isotopy in an oriented 3-manifold is when the manifold admits a properly embedded non-separating 2-spheres. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of 3-manifolds. We also express our results in the language of skein modules. In particular, we relate our results to the presence of torsion in the framing skein module.

For a finite-type surface S, we study a preferred basis for the commutative algebra C[R_SL3(S)] of regular functions on the SL3(C)-character variety, introduced by Sikora-Westbury. These basis elements come from the trace functions associated to certain tri-valent graphs embedded in the surface S. We show that this basis can be naturally indexed by positive integer coordinates, defined by Knutson-Tao rhombus inequalities and modulo 3 congruence conditions. These coordinates are related, by the geometric theory of Fock-Goncharov, to the tropical points at infinity of the dual version of the character variety. This is joint work with Zhe Sun.

The combinatorial quantization of character varieties of surfaces (a.k.a moduli space of flat connections) has been introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid-90's. In this talk, I will define the quantized analogue of the holonomy of a flat connection and explain its properties. Then I will use holonomy to relate combinatorial quantization with certain stated skein algebras and obtain finite presentations of the latter.

Khovanov homology, an invariant of knots and links, is a categorification of the Jones polynomial. $\mathbb{Z}_2$ torsion is very common in Khovanov homology. However, other torsion groups appear rarely.

In this talk, I will focus on recent developments regarding non-$\mathbb{Z}_2$ torsion in Khovanov homology. After discussing some infinite families of knots and links with non-$\mathbb{Z}_2} torsion, I will establish that torsion in Khovanov homology can be arbitrarily large. To conclude, I will mention some current research and open questions.

The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction of link invariants. In this context, the quantum group Uq(sl(2)) leads to the sequence of coloured Jones polynomials, which contains the original Jones polynomial. Dually, the quantum group at roots of unity gives the sequence of coloured Alexander polynomials. We construct a unified topological model for these two sequences of quantum invariants. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the Nth coloured Jones and Nth coloured Alexander invariants come as different specialisations of a state sum (defined over 3 variables) of Lagrangian intersections in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space.

Given a fusion rule $q\otimes q \cong 1\oplus\bigoplus{i=1}^{k} x_{i}$ in a unitary fusion category C, we extract information using skein-theoretic methods and a rotation operator. For instance, one can classify all associated framed link invariants when k = 1, 2 and C is ribbon. We consider an action of the rotation operator on a “canonical basis”. Assuming self-duality of the summands $x_i$ , we will explore some properties of certain 6j-symbols using skein theory. We also explore some features of quantum invariants coming from q antisymmetrically self-dual. This is joint work with Sachin J. Valera.

In a series of joint works with Tian Yang, we made a volume conjecture and an asymptotic expansion conjecture for the relative Reshetikhin-Turaev invariants for a closed oriented 3-manifold with a colored framed link inside it. We propose that their asymptotic behavior is related to the volume, the Chern-Simons invariant and the adjoint twisted Reidemeister torsion associated with the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring.

In this talk, I will first discuss how our volume conjecture can be understood as an interpolation between the Kashaev-Murakami-Murakami volume conjecture of the colored Jones polynomials and the Chen-Yang volume conjecture of the Reshetikhin-Turaev invariants. Then I will describe how the adjoint twisted Reidemeister torsion shows up in the asymptotic expansion of the invariants. Especially, we find new explicit formulas for the adjoint twisted Reidemeister torsion for the fundamental shadow link complements and for the 3-manifold obtained by doing hyperbolic Dehn-filling on those link complements. Those formulas cover a very large class of hyperbolic 3-manifold and appear naturally in the asymptotic expansion of quantum invariants. Finally, I will summarize the recent progress of the asymptotic expansion conjecture for the fundamental shadow link pairs.