$\mathrm{SL}_2(\mathbb{C})$-holonomy invariants of links

Abstract

Quantum invariants like the colored Jones polynomial are algebraic in nature but are conjectured to detect important information about the geometry of links. In this thesis we explore these connections using an enhanced version of the RT construction. Our invariants take the holonomy of a flat connection on the link complement as input, so we call them holonomy invariants. The case of trivial holonomy recovers the ordinary RT construction. We consider holonomy representations into $\operatorname{SL}_2(\mathbb C)$, which are closely related to hyperbolic geometry. In order to define our invariants we consider a particular coordinate system on the space of representations closely related to the octahedral decomposition of the knot complement. We call the corresponding diagrams shaped tangles. Using shaped tangles we define a family of holonomy invariants $\mathrm{J}_N$ indexed by integers $N \ge 2$, which we call the nonabelian quantum dilogarithm. They can be interpreted as a noncommutative deformation of Kashaev's quantum dilogarithm (equivalently, the $N$th colored Jones polynomial at a $N$th root of unity) or of the ADO invariants, depending on the eigenvalues of the holonomy. Our construction depends in an essential way on representations of quantum $\mathfrak{sl}_2$ at $q = \xi$ a primitive $2N$th root of unity. We show that $\mathrm{J}_N$ is defined up to a power of $\xi$ and does not depend on the gauge class of the holonomy. Afterwards we introduce a version of the quantum double construction for the holonomy invariants. We show that the quantum double $\mathrm{T}_N$ of the nonabelian dilogarithm $\mathrm{J}_N$ admits a canonical normalization with no phase ambiguity. Finally, we prove that in the case $N = 2$ the doubled invariant $\mathrm{T}_2$ computes the Reidemeister torsion of the link complement twisted by the holonomy representation.