arXiv subjects

Stavros Garoufalidis

Publications and source records attributed to Stavros Garoufalidis.

At least 37 records · Page 2Linked to original sources

arXiv2024

Knots, Perturbative Series and Quantum Modularity

We introduce an invariant of a hyperbolic knot which is a map $α\mapsto \boldsymbolΦ_α(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundary parabolic ${\rm SL}_2(\mathbb{C})$ representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their $(σ_0,σ_1)$ entry, where $σ_0$ is the trivial and $σ_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\rm e}^{2π{\rm i} α}$ as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; (c) the columns of $\boldsymbolΦ$ are fundamental solutions of a linear $q$-difference equation; (d) the matrix defines an ${\rm SL}_2(\mathbb{Z})$-cocycle $W_γ$ in matrix-valued functions on $\mathbb{Q}$ that conjecturally extends to a smooth function on $\mathbb{R}$ and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series $\boldsymbolΦ(h)$ to actual functions. The two invariants $\boldsymbolΦ$ and $W_γ$ are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent $q$-series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.

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arXiv2024

The 3d-index of the 3d-skein module via the quantum trace map

We define a map from the skein module of a cusped hyperbolic 3-manifold to the ring of Laurent series in one variable with integer coefficients that satisfies two properties: its evaluation at peripheral curves coincides with the Dimofte--Gaiotto--Gukov 3d-index, and it factors through the 3d-quantum trace map associated to a suitable ideal triangulation of the manifold. The map fulfills a supersymmetry prediction of mathematical physics and is part of a conjectural 3+1 dimensional topological quantum field theory.

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arXiv2024

Multivariable knot polynomials from braided Hopf algebras with automorphisms

We construct knot invariants from solutions to the Yang--Baxter equation associated to appropriately generalized left/right Yetter--Drinfel'd modules over a braided Hopf algebra with an automorphism. When applied to Nichols algebras, our method reproduces known knot polynomials and naturally produces multivariable polynomial invariants of knots. We discuss in detail Nichols algebras of rank $1$ which recover the ADO and the colored Jones polynomials of a knot and two sequences of examples of rank $2$ Nichols algebras, one of which starts with the product of two Alexander polynomials, and then conjecturally the Harper polynomial. The second sequence starts with the Links--Gould invariant (conjecturally), and then with a new 2-variable knot polynomial that detects chirality and mutation, and whose degree gives sharp bounds for the genus for a sample of 30 computed knots.

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arXiv2024

On the trace fields of hyperbolic Dehn fillings

Assuming Lehmer's conjecture, we estimate the degree of the trace field $K(M_{p/q})$ of a hyperbolic Dehn-filling $M_{p/q}$ of a 1-cusped hyperbolic 3-manifold $M$ by $$ \dfrac{1}{C}(\max\;\{|p|,|q|\})\leq \text{deg }K(M_{p/q}) \leq C(\max\;\{|p|,|q|\}) $$ where $C=C_M$ is a constant that depends on $M$.

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arXiv2024

A Quantum trace map for 3-manifolds

We define a quantum trace map from the skein module of a 3-manifold with torus boundary components to a module (left and right quotient of a quantum torus) constructed from an ideal triangulation. Our map is a 3-dimensional version of the well-known quantum trace map on surfaces introduced by Bonahon and Wong and further developed by Lê.

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arXiv2024

Super-representations of 3-manifolds and torsion polynomials

Torsion polynomials connect the genus of a hyperbolic knot (a topological invariant) with the discrete faithful representation (a geometric invariant). Using a new combinatorial structure of an ideal triangulation of a 3-manifold that involves edges as well as faces, we associate a polynomial to a cusped hyperbolic manifold that conjecturally agrees with the $\BC^2$-torsion polynomial, which conjecturally detects the genus of the knot. The new combinatorics is motivated by super-geometry in dimension 3, and more precisely by super-Ptolemy assignments of ideally triangulated 3-manifolds and their $\mathrm{OSp}_{2|1}(\BC)$-representations. Extended section 4, and added superalgebras with a single odd generator.

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arXiv2024

The (twisted/$L^2$)-Alexander polynomial of ideally triangulated 3-manifolds

We establish a connection between the Alexander polynomial of a knot and its twisted and $L^2$-versions with the triangulations that appear in 3-dimensional hyperbolic geometry. Specifically, we introduce twisted Neumann--Zagier matrices of ordered ideal triangulations and use them to provide formulas for the Alexander polynomial and its variants, the twisted Alexander polynomial and the $L^2$-Alexander torsion.

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arXiv2023

Knots and Their Related $q$-Series

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.

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arXiv2023

From 3-dimensional skein theory to functions near Q

Motivated by the Quantum Modularity Conjecture and its arithmetic aspects related to the Habiro ring of a number field, we define a map from the Kauffman bracket skein module of an integer homology 3-sphere to the Habiro ring, and use Witten's conjecture (now a theorem) to show that the image is an effectively computable module of finite rank that can be used to phrase the quantum modularity conjecture.

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arXiv2023

Perturbative invariants of cusped hyperbolic 3-manifolds

We prove that a formal power series associated to an ideally triangulated cusped hyperbolic 3-manifold (together with some further choices) is a topological invariant. This formal power series is conjectured to agree to all orders in perturbation theory with two important topological invariants of hyperbolic knots, namely the Kashaev invariant and the Andersen--Kashaev invariant (also known as the state-integral) of Teichmüller TQFT.

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arXiv2023

Asymptotically multiplicative quantum invariants

The Euler characteristic and the volume are two best-known multiplicative invariants of manifolds under finite covers. On the other hand, quantum invariants of 3-manifolds are not multiplicative. We show that a perturbative power series, introduced by Dimofte and the first author and shown to be a topological invariant of cusped hyperbolic 3-manifolds by Storzer--Wheeler and the first author, and conjectured to agree with the asymptotics of the Kashaev invariant to all orders in perturbation theory, is asymptotically multiplicative under cyclic covers. Moreover, its coefficients are determined by polynomials constructed out of twisted Neumann--Zagier data. This gives a new $t$-deformation of the perturbative quantum invariants, different than the $x$-deformation obtained by deforming the geometric representation. We illustrate our results with several hyperbolic knots.

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arXiv2023

Quantum dilogarithms over local fields and invariants of 3-manifolds

To each local field (including the real or complex numbers) we associate a quantum dilogarithm and show that it satisfies a pentagon identity and some symmetries. Using an angled version of these quantum dilogarithms, we construct three generalized TQFTs in 2+1 dimensions, one given by a face state-integral and two given by edge state-integrals. Their partition functions rise to distributional invariants of 3-manifolds with torus boundary, conjecturally related to point counting of the $A$-polynomial curve. The partition function of one of these face generalized TQFTs for the case of the real numbers can be expressed either as a multidimensional Barnes-Mellin integral or as a period on a curve which is conjecturally the $A$-polynomial curve.

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arXiv2023

Hyperbolic 3-manifolds, the Bloch group, and the work of Walter Neumann

This is an article about the work of Walter Neumann on hyperbolic geometry, ideal triangulations of 3-manifolds, the volume and Chern-Simons invariants of 3-manifolds and their elements of the the Bloch group. The article focuses on the relations of these classical invariants and the impact of Neumann's work to quantum topology and to mathematical physics.

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arXiv2022

The descendants of the 3d-index

In the study of 3d-3d correspondence occurs a natural $q$-Weyl algebra associated to an ideal triangulation of a 3-manifold with torus boundary components, and a module of it. We study the action of this module on the (rotated) 3d-index of Dimofte-Gaiotto-Gukov and we conjecture some structural properties: bilinear factorization in terms of holomorphic blocks, pair of linear $q$-difference equations, the determination of the 3d-index in terms of a finite size matrix of rational functions and the asymptotic expansion of the $q$-series as $q$ tends to 1 to all orders. We illustrate our conjectures with computations for the case of the three simplest hyperbolic knots.

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arXiv2022

Factorization of polynomials in hyperbolic geometry and dynamics

Using factorization theorems for sparse polynomials, we compute the trace field of Dehn fillings of the Whitehead link, and (assuming Lehmer's Conjecture) the minimal polynomial of the small dilatation pseudo-Anosov maps and the trace field of fillings of the figure-8 knot. These results depend on the degrees of the trace fields over Q being sufficiently large.

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arXiv2022

Periods, the meromorphic 3D-index and the Turaev--Viro invariant

The 3D-index of Dimofte-Gaiotto-Gukov is an interesting collection of $q$-series with integer coefficients parametrised by a pair of integers and associated to a 3-manifold with torus boundary. In this note, we explain the structure of the asymptotic expansions of the 3D-index when $q=e^{2πiτ}$ and $τ$ tends to zero (to all orders and with exponentially small terms included), and discover two phenomena: (a) when $τ$ tends to zero on a ray near the positive real axis, the horizontal asymptotics of the meromorphic 3D-index match to all orders with the asymptotics of the Turaev-Viro invariant of a knot, in particular explaining the Volume Conjecture of Chen-Yang from first principles, (b) when $τ\to 0$ on the positive imaginary axis, the vertical asymptotics of the 3D-index involves periods of a plane curve (the $A$-polynomial), as opposed to algebraic numbers, explaining some predictions of Hodgson-Kricker-Siejakowski and leading to conjectural identities between periods of the $A$-polynomial of a knot and integrals of the Euler beta-function.

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arXiv2022

The descendant colored Jones polynomials

We discuss two realizations of the colored Jones polynomials of a knot, one from an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another one from recent work of D. Zagier and the first author regarding the Refined Quantum Modularity Conjecture, more precisely, the mysterious top row of a matrix of conjectured knot invariants.

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arXiv2022

Peacock patterns and resurgence in complex Chern-Simons theory

The partition function of complex Chern-Simons theory on a 3-manifold with torus boundary reduces to a finite dimensional state-integral which is a holomorphic function of a complexified Planck's constant $τ$ in the complex cut plane and an entire function of a complex parameter $u$. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear $q$-difference equation. We further conjecture that entries of the Stokes automorphism matrix are the 3D-indices of Dimofte-Gaiotto-Gukov. We provide proofs of our statements regarding the $q$-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic $4_1$ and $5_2$ knots.

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