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Calvin McPhail-Snyder

Publications and source records attributed to Calvin McPhail-Snyder.

11 recordsLinked to original sources
arXiv2026

Zesting and the relative complexity of Reshetikhin-Turaev invariants

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $ζ$ to produce a new category $\mathcal{C}^ζ$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_ζ$ depending only on $ζ$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute \emph{tangle} invariants $\mathcal{J}_ζ(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_ζ(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_ζ(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.

math.QA↗
arXiv2026

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

We define a sequence of invariants $\mathcal{Z}_{N}^ψ$ 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 $\mathcal{I}^ψ$ of the Chern-Simons invariant of a tangle exterior. $\mathcal{Z}_{N}^ψ$ directly recovers $\mathcal{I}^ψ$ when $N = 1$. We build $\mathcal{Z}_{N}^ψ$ 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 $\mathcal{Z}_{N}^ψ$ is defined without any phase ambiguity. It is natural to conjecture that $\mathcal{Z}_{N}^ψ$ 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.

math.QA↗
arXiv2026

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

Previous work of the author and N. Reshetikhin defines an invariant $\operatorname{Z}_{N}^ψ(K, ρ, μ)$ of a knot $K$, a representation $ρ: π_{1}(S^{3} \setminus K) \to \operatorname{SL}_2(\mathbb{C})$, and a logarithm $μ$ of a meridian eigenvalue of $ρ$. It can be interpreted as a geometric twist of the Kasahev invariant or as a quantization of the $\operatorname{SL}(\mathbb{C})$ Chern-Simons invariant and is defined using a discrete state-sum involving quantum dilogarithms. In this paper we show how to express $\operatorname{Z}_{N}^ψ(K, ρ, μ)$ as a sum over contour integrals in a space parametrizing hyperbolic structures on the knot complement. Such integral presentations are an important step in determining the asymptotics of quantum invariants as predicted by the Volume Conjecture. We discuss this perspective and the remaining obstacles to establishing exponential growth of $\operatorname{Z}_{N}^ψ$.

math.GT↗
arXiv2026

Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$

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 $\mathcal{U}_ξ(\mathfrak{sl}_2)$ at a root of unity $ξ$. 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.

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arXiv2025

Octahedral coordinates from the Wirtinger presentation

Let $ρ$ 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 $D$. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. $ρ$ induces a hyperbolic structure on the complement of $D$, and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether $ρ$ occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function.

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arXiv2022

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

Classical $\operatorname{SL}_2(\mathbb{C})$-Chern-Simons theory assigns a $3$-manifold $M$ with representation $ρ: π_1(M) \to \operatorname{SL}_2(\mathbb{C})$ its complex volume $\operatorname{V}(M, ρ) \in \mathbb{C} / 2 π^2 i \mathbb{Z}$, with real part the volume and imaginary part the Chern-Simons invariant. The existing literature focuses on computing $\operatorname{V}$ using a triangulation. In this paper we show how to compute $\operatorname{V}(M, L, ρ)$ directly from a surgery diagram for $M$ a compact oriented $3$-manifold with torus boundary components, embedded cusps $L$, and representation $ρ: π_1(M \setminus L) \to \operatorname{SL}_2(\mathbb{C})$. When $M$ has nonempty boundary $\operatorname{V}(M, L, ρ)(\mathfrak{s})$ depends on some extra data $\mathfrak{s}$ we call a log-decoration. Our method describes $ρ$ in a coordinate system closely related to quantum groups, and we think of our construction as a classical, noncompact version of Witten-Reshetikhin-Turaev's quantum $\operatorname{SU}(2)$ Chern-Simons theory.

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arXiv2021

Kashaev--Reshetikhin Invariants of Links

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 $\mathrm{KaRe}(K)$ 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.

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arXiv2021

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

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 $q = ξ$ a primitive $2N$th root of unity. We show that $\mathrm{J}_N$ 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 $\mathrm{T}_N$ 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.

math.QA↗
arXiv2021

Holonomy invariants of links and nonabelian Reidemeister torsion

We show that the reduced $\mathrm{SL}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $\mathcal{U}_q(\mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $\operatorname{SL}_2(\mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.

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arXiv2018

Planar diagrams for local invariants of graphs in surfaces

In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, the $S$-polynomial, and formulate the $\mathfrak{sl}(N)$ Penrose polynomial for non-cubic graphs, giving contraction-deletion relations. The $S$-polynomial is used to define an extension of the Yamada polynomial to virtual spatial graphs, and with it we obtain a sufficient condition for non-classicality of virtual spatial graphs. We conjecture the existence of local relations for the $S$-polynomial at squares of integers.

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