Recently, Kashaev and the first author defined a sequence \(V_n\) of \(2\)-variable knot polynomials with integer coefficients, coming from the \(R\)-matrix of a rank \(2\) Nichols algebra, the first polynomial been identified with the Links-Gould polynomial. In this note we present the results of the computation of the \(V_n\) polynomials for \(n=1,2,3,4\) and discover applications and emerging patterns, including unexpected Conway mutations that seem undetected by the \(V_n\)-polynomials as well as by Heegaard Floer Homology and Knot Floer Homology.

Matrix-valued holomorphic quantum modular forms are intricate objects that arise in successive refinements of the Volume Conjecture of knots and involve three holomorphic, asymptotic and arithmetic objects. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the \((-2,3,7)\)-pretzel knot.

Quantum knot invariants are known to come from $R$-matrices along with some extra structures, a process called the Reshetikhin--Turaev functor. In $2019$, Rinat Kashaev proved that $R$-matrices alone are sufficient to define knot invariants, as long as they satisfy some nondegeneracy conditions called rigidity. More recently, Stavros Garoufalidis and Rinat Kashaev developed a new method of constructing rigid $R$-matrices, which recovers several known knot polynomials such as the colored Jones polynomials, and gives a new family of multivariable knot polynomials, the $V_n$-polynomials. In this talk, I will talk about the Reshetikhin--Turaev functor in this context, the computation of $V_n$-polynomials and the patterns of the results based on $361404$ knots computed. Joint work with Stavros Garoufalidis.

Quantum topology is considered to be initiated by the discovery of the Jones polynomial in 1984, followed with observations of numerous links to physics. In the late '80s, Atiyah, Segal, and Witten established an intrinsic definition of the Jones polynomial using \(\text{SU}(2)\) Chern-Simons theory, revealing the rich connections of the Jones polynomial with the physical world. Successive findings around the Jones polynomial emerged, including one famous conjecture that is the main topic of this thesis, the Quantum Modularity Conjecture.

In 1995, R. Kashaev introduced a knot invariant using the quantum dilogarithm function, which for a hyperbolic knot K is conjectured to have an exponential growth rate, a conjecture known as the Volume Conjecture. In 2001, H. Murakami and J. Murakami discovered that Kashaev's invariant is equal to the value of the \(N\)-colored Jones polynomial at \(N\)-th roots of the unity. With this, D. Zagier observed a modular relation between the values of the \(N\)-colored Jones polynomial at different roots of the unity and extended the statement of Volume Conjecture to a modular relation of the functions. The extended statement is known as the Quantum Modularity Conjecture (QMC).

More recently, J. E. Andersen and R. Kashaev introduced the Teichmuller TQFT based on Chern-Simons theory with infinite dimensional gauge groups, promoting the quantum Teichmuller theory to a TQFT of categroids. On further investigation into values of the Teichmuller TQFT on knot complements of the \(4_1\) knot and the \(5_2\) knot, S. Garoufalidis and D. Zagier discovered phenomena suggesting deep relationships between the state integral from the Teichmuller TQFT and QMC. Furthermore, their observation also suggested rich connections with several other topics, such as the Dimofte-Gaiotto-Gukov index and the quantum spin network.

This thesis will mainly focus on introducing the construction of the Teichmuller TQFT and the contents of QMC, and demonstrate their connections by listing the observations made by S. Garoufalidis and D. Zagier and more recent results along with elementary proofs for some of them from joint work of the author, N. An and S. Garoufalidis.

Supergeometry is a natural extension of the theory of differential geometry, which enjoys values on its own right as a purely mathematical object, and also turns out to be useful in physics: it gives a model of spacetime that unifies quantum science and gravity, the string theory. The first part of this thesis gives a detailed and mathematically strict introduction to supergeometry, rearranged from the lecture notes by Covolo and Poncin, the paper of Leites and the notes by Deligne and Morgan. The second part focuses on an explicit discussion on the important example \(\underline{\text{SMan}}(R^{0|\delta},X)\) and outlines a corresponding proof of the Chern-Gauss-Bonnet theorem, following Berwick-Evans’ work.