UC Berkeley

The Reshetikhin-Turaev construction is a method of obtaining invariants of links (and other topological objects) via the representation theory of quantum groups.It underlies quantum invariants such as the Jones polynomial and its many generalizations. These invariants 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.

The geometry of a link complement can be described by a representation of its fundamental group into a Lie group, equivalently the holonomy of a flat Lie algebra-valued connection.Our invariants take this data 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 in terms of diagrams we call shaped tangles. We show that these coordinates are closely related to the shape parameters of a certain ideal triangulation (the octahedral decomposition) of the link complement.

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 $r = 2$ the doubled invariant $\mathrm{T}_2$ computes the Reidemeister torsion of the link complement twisted by the holonomy representation.