# Papers

- Octahedral coordinates from the Wirtinger presentation
- submitted

arXiv: 2404.19155 [math.GT] ## Summary

Let \(\rho\) be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into \(\operatorname{SL}_2(\mathbb{C})\) expressed in terms of the Wirtinger generators of a diagram \(D\) . In this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition of the knot complement associated to \(D\) . Our formula gives a new, explicit criterion for whether \(\rho\) occurs as a critical point of the diagram’s Neumann-Zagier–Yokota potential function.

- Hyperbolic structures on link complements, octahedral decompositions, and quantum \(\mathfrak{sl}_2\)
- submitted

arXiv: 2203.06042 [math.GT] ## Summary

Hyperbolic structures (equivalently, principal \(\operatorname{PSL}_2(\mathbb C)\) -bundles with connection) on link complements can be described algebraically by using the

*octahedral decomposition*, which assigns an ideal triangulation to any diagram of the link. The decomposition (like any ideal triangulation) gives a set of*gluing equations*in*shape parameters*whose solutions are hyperbolic structures. We show that these equations are closely related to a certain presentation of the*Kac-de Concini quantum group*\(\mathcal{U}_q(\mathfrak{sl}_2)\) in terms of cluster algebras at \(q = \xi\) a root of unity. Specifically, we identify ratios of the shape parameters of the octahedral decomposition with central characters of \(\mathcal{U}_\xi(\mathfrak{sl}_2)\) . The quantum braiding on these characters is known to be closely related to \(\operatorname{SL}_2(\mathbb C)\) -bundles on link complements, and our work provides a geometric perspective on this construction.- Kashaev-Reshetikhin invariants of links
- submitted

arXiv: 2108.06561 [math.GT] ## Summary

Kashaev and Reshetikhin previously described a way to define holonomy invariants of knots using quantum \(\mathfrak{sl}_2\) at a root of unity. These are generalized quantum invariants depend both on a knot \(K\) and a representation of the fundamental group of its complement into \(\mathrm{SL}_2(\mathbb{C})\) ; equivalently, we can think of \(\mathrm{KR}(K)\) as associating to each knot a function on (a slight generalization of) its character variety. In this paper we clarify some details of their construction. In particular, we show that for \(K\) a hyperbolic knot \(\mathrm{KR}(K)\) can be viewed as a function on the geometric component of the \(A\) -polynomial curve of \(K\) . We compute some examples at a third root of unity.

- Holonomy invariants of links and nonabelian Reidemeister torsion
- in
Quantum Topology

arXiv: 2005.01133 [math.QA]

doi: 10.4171/QT/160 ## Summary

We show that the \(\mathrm{SL}_2(\mathbb{C})\) -twisted Reidemeister torsion of a link can be computed using quantum \(\mathfrak{sl}_2\) at a fourth root of unity. The proof uses a Schur-Weyl duality with the Burau representation. Our construction is a special case of the

*quantum holonomy invariants*of Blanchet, Geer, Patureau-Mirand, and Reshetikhin and we consequently interpret their invariant as a twisted Conway potential.- Planar diagrams for local invariants of graphs in surfaces
- in
Journal of Knot Theory and its Ramifications

arXiv: 1805.00575 [math.GT]

doi: 10.1142/S0218216519500937 ## Summary

In order to apply quantum topology methods to non-planar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. We also discuss an extension of the flow polynomial called the \(S\) -polynomial and relate it to the Yamada and Penrose polynomials.

- Surgery calculus for classical \(\operatorname{SL}_2(\mathbb{C})\) Chern−Simons theory
- arXiv: 2210.09469 [math.GT]
## Summary

In this article I work out how to compute complex volumes of link complements directly from their diagrams using the octahedral decomposition. This is an application of the results of arXiv:2203.06042. I also show how to extend this computation to general \(3\) -manifolds via surgery; this is a sort of classical version of the surgery calculus for the Witten-Reshetikhin-Turaev theory. For link complements the complex volume depends on an extra choice of boundary data called a

*log-decoration*; this was known to experts but usually not discussed explicitly in the literature. Most of the results in this paper are already known in some form; my main goal was to explain them in a unified way. I plan on publishing an expanded version with some new material.- \(\mathrm{SL}_2(\mathbb{C})\)-holonomy invariants of links
- PhD thesis

arXiv: 2105.05030 [math.QA] ## Summary

My PhD thesis gives an improved version of the \(\mathrm{SL}_2(\mathbb{C})\) -holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin. Among other things, we describe a coordinate system for \(\mathrm{SL}_2(\mathbb{C})\) -tangles that is directly related to hyperbolic geometry (via the octahedral decomposition of the complement), explicitly compute the braiding matrices, and reduce the scalar ambiguity to a \(2N\) th root of unity. We also describe the quantum double construction for holonomy invariants and give the relationship with the torsion (as first published in

*Holonomy invariants of links and nonabelian Reidemeister torsion*) in this context.