2021/05/11 by Calvin McPhail-Snyder, McPhail-Snyder, Calvin
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Combinatorics #Geometric and Algebraic Topology #Holonomy #Mathematics #Physics #Pure mathematics #math.GT #math.QA
paper · pdf · doi:10.48550/arxiv.2105.05030
published in arXiv (Cornell University) (Cornell University) · PhD thesis. Extends arXiv:2005.01133
arxiv created 2021/05/11 · openalex publication_date 2021/05/11 · arxiv updated 2021/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 SL2(\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 JN indexed by integers N ≥ 2, which we call the nonabelian quantum dilogarithm. They can be interpreted as a noncommutative deformation of Kashaev's quantum dilogarithm (equivalently, the Nth colored Jones polynomial at a Nth 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 \mathfraksl2 at q = ξ a primitive 2Nth root of unity. We show that JN is defined up to a power of ξ 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 TN of the nonabelian dilogarithm JN admits a canonical normalization with no phase ambiguity. Finally, we prove that in the case N = 2 the doubled invariant T2 computes the Reidemeister torsion of the link complement twisted by the holonomy representation.