Papers

Research papers

State integrals for the quantized \(\operatorname{SL}_2(\mathbb{C})\) Chern-Simons invariant

arXiv: 2601.05136 [math.GT]

Summary

The geometric quantum invariants \(\operatorname{Z}_{N}^{\psi}(K, \rho, \mu)\) of [arXiv:2509.02365] are defined using discrete state sums of quantum dilogarithms. This paper shows how to re-write them as a sum of state integrals in a space parametrizing hyperbolic structures on \(S^3 \setminus K\) . Unlike previous similar results this presentation is exact (not asymptotic) and exists for any sufficiently nondegenerate representation. Such state integrals appear in the usual approach to the Volume Conjecture. We discuss this perspective and barriers to proving asymptotic exponential growth of \(\operatorname{Z}_{N}^{\psi}(K, \rho, \mu)\) .

A quantization of the \(\operatorname{SL}_2(\mathbb{C})\) Chern-Simons invariant of tangle exteriors

submitted
arXiv: 2509.02365 [math.QA]

Summary

We define a sequence of invariants \(\mathcal{Z}_{N}^{\psi}\) of tangles with flat \(\mathfrak{sl}_{2}\) connections (i.e. hyperbolic structures) on their complements. These can be interpreted as a geometric twist of the Kashaev invariant or as a quantization of the \(\operatorname{SL}_{2}(\mathbb{C})\) Chern-Simons invariant. To support the second interpretation we give a new description \(\mathcal{I}^{\psi}\) of the Chern-Simons invariant of a tangle exterior. \(\mathcal{Z}_{N}^{\psi}\) directly recovers \(\mathcal{I}^{\psi}\) when \(N = 1\) . We build \(\mathcal{Z}_{N}^{\psi}\) using modules over unrestricted quantum \(\mathfrak{sl}_{2}\) at a root of unity and the holonomy \(R\) -matrices previously constructed by the author and Reshetikhin [arXiv:2509.02354]. Unlike most previous constructions of geometric quantum invariants \(\mathcal{Z}_{N}^{\psi}\) is defined without any phase ambiguity. It is natural to conjecture that \(\mathcal{Z}_{N}^{\psi}\) is related to the quantization of Chern-Simons theory with complex, noncompact gauge group \(\operatorname{SL}_{2}(\mathbb{C})\) and we discuss how to interpret our results in this context.

The holonomy braiding for \(\mathcal{U}_\xi(\mathfrak{sl}_2)\) in terms of geometric quantum dilogarithms

with Nicolai Reshetikhin

submitted
arXiv: 2509.02354 [math.QA]

Summary

We derive an explicit formula for the holonomy \(R\) -matrix of quantum \(\mathfrak{sl}_2\) at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.

Octahedral coordinates from the Wirtinger presentation

in Geometriae Dedicata
arXiv: 2404.19155 [math.GT]
doi: 10.1007/s10711-025-01012-7

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 on link complements (equivalently, representations of the fundamental group into \(\operatorname{SL}_2(\mathbb{C})\) ) can be described algebraically by using the octahedral decomposition determined by a link diagram. 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 can be obtained from Kashaev-Reshetikhin’s braiding on the Kac-de Concini quantum group \(\mathcal{U}_\xi(\mathfrak{sl}_2)\) at a root of unity \(\xi\) . This braiding gives coordinates on the \(\operatorname{SL}_2(\mathbb{C})\) representation variety of a link and our work shows how to interpret these geometrically.

Kashaev-Reshetikhin invariants of links

with Kai-Chieh Chen, Scott Morrison, and Noah Snyder

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

with Kyle Miller

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.

Other writing

Surgery calculus for classical \(\operatorname{SL}_2(\mathbb{C})\) Chern−Simons theory

arXiv: 2210.09469 [math.GT]

Summary

In this note 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 content is also contained in the later article arXiv:2509.02365 which extends these ideas to define a Chern-Simons invariant for tangles.

\(\mathrm{SL}_2(\mathbb{C})\)-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.