arXiv subjects

Stavros Garoufalidis

Publications and source records attributed to Stavros Garoufalidis.

At least 19 recordsLinked to original sources

arXiv2026

The $\mathrm{FK}_3$ knot polynomial

We present the knot polynomial associated to the 12-dimensional sporadic Nichols algebra $\mathrm{FK}_3$ with automorphism, compute it for all knots with $\leq$ 16 crossings and discuss the found patterns. This knot polynomial is not a Vassiliev power series. We ponder how it compares to the knot polynomials that come from Lie algebras and their representations and what the other knot polynomials associated with the higher dimensional sporadic Nichols algebras are.

math.GT↗

arXiv2026

Genus bounds for knot polynomials of Lie superalgebras

Knot polynomials colored by typical representations of Lie superalgebras of type I (except $\mathfrak{psl}(n|n)$) have two variables $q$ and $t$, the latter corresponding to the complex-valued weight of the distinguished odd root. We prove that for every typical representation of a Lie superalgebra of type I, the $t$-degree of the knot polynomial is at most the number of odd roots times the genus of the knot. A complimentary bound being at least the number of odd roots times degree of the Alexander polynomial can be obtained from a specialization at $q=1$. These two bounds become equalities when the Alexander polynomial detects the genus of the knot, as is the case for alternating knots and fibered knots.

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arXiv2026

Positivity and log concavity of the Links--Gould polynomial of knots

Motivated by the recent work of Harper--Kohli--Song--Tahar, we formulate a positivity, hole-free, and log-concavity conjecture for the Links--Gould polynomial of alternating links and verify it for all 51.3 million alternating knots with at most 19 crossings. All but 544 of those knots satisfy a stronger type-B log-concavity condition characterized by the slopes of edges in the subdivision of the monomial support induced by the log coefficients.

math.GT↗

arXiv2026

Combinatorial description of closed $3$-manifolds via ordered ideal triangulations

It is well known that every compact oriented 3-manifold admits an ideal triangulation, and that any two such triangulations with at least two ideal tetrahedra are related by a sequence of Pachner $2$-$3$ moves. Motivated by constructions in quantum topology, we give a combinatorial description of closed $3$-manifolds in terms of ordered ideal triangulations and ordered Pachner $2$-$3$ and $0$-$2$ moves.

math.GT↗

arXiv2026

Extending Knot Polynomials of Braided Hopf Algebras to Links

Recently, a plethora of multivariable knot polynomials were introduced by Kashaev and one of the authors, by applying the Reshetikhin-Turaev functor to rigid $R$-matrices that come from braided Hopf algebras with automorphisms. We study the extension of these knot invariants to links, and use this to identify some of them with known link invariants, as conjectured in that same recent work.

math.QA↗

arXiv2026

Resurgence of Chern-Simons theory at the trivial flat connection

Some years ago, it was conjectured by the first author that the Chern-Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series (including the location of the singularities and their Stokes constants) in the case of a hyperbolic knot complement in terms of an extended square matrix of $(x,q)$-series whose rows are indexed by the boundary parabolic $\text{SL}_2(\mathbb{C})$-flat connections, including the trivial one. We use our extended matrix to describe the Stokes constants of the above series, to define explicitly their Borel transform and to identify it with state-integrals. Along the way, we use our matrix to give an analytic extension of the Kashaev invariant and of the colored Jones polynomial and to complete the matrix valued holomorphic quantum modular forms as well as to give an exact version of the refined quantum modularity conjecture of Zagier and the first author. Finally, our matrix provides an extension of the 3D-index in a sector of the trivial flat connection. We illustrate our definitions, theorems, numerical calculations and conjectures with the two simplest hyperbolic knots.

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arXiv2026

Patterns of the $V_2$-polynomial of knots

Recently, Kashaev and the first author constructed an $R$-matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin--Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence $V_n$ of 2-variable knot polynomials with integer coefficients, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the $V_n$-polynomials for $n=1,2,3,4$. This leads to the discovery of emerging patterns, including the genus bound for $V_2$ being an equality for all 352.2 million knots with at most $19$ crossings, as well as unexpected Conway mutations that seem undetected by the $V_n$-polynomials as well as by Heegaard Floer Homology and Khovanov Homology.

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arXiv2026

Skein theory for the Links-Gould polynomial

Building further on work of Marin and Wagner, we give a cubic braid-type skein theory of the Links--Gould polynomial invariant of oriented links and prove that it can be used to evaluate any oriented link, adding this polynomial to the list of polynomial invariants that can be computed by skein theory. As a consequence, we prove that this skein theory is also shared by the $V_1$-polynomial defined by two of the authors, deducing the equality of the two link polynomials. This implies specialization properties of the $V_1$-polynomial to the Alexander polynomial and to the $\mathrm{ADO}_3$-invariant, the fact that it is a Vassiliev power series invariant, as well as a Seifert genus bound for knots.

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arXiv2026

A lift of the colored Jones polynomial of a knot

Habiro lifted the Witten-Reshetikhin-Turaev invariant of an integer homology 3-sphere (a complex-valued function on the set of complex roots of unity) to an element of the Habiro ring. We lift the colored Jones polynomial of a knot, with Alexander polynomial $Δ(t)$, to the recently introduced Habiro ring of the étale map $\mathbb{Z}[t^{\pm 1}]\to \mathbb{Z}[t^{\pm 1},Δ(t)^{-1}]$ (with Frobenius lifts $t\mapsto t^p$ for all primes $p$). This implies the existence of a loop expansion at roots of unity (confirming a conjecture of Habiro), and a lift of power series invariants of Ohtsuki for 3-manifolds with Betti number 1 to a Habiro ring. Our results have natural extensions to the skein module of a knot complement, and they suggest a natural lift of the colored Jones polynomial colored by representations of a simple Lie (super) algebra.

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arXiv2026

A relation between the Baseilhac-Benedetti and the Bonahon-Liu-Wong-Yang invariants

Baseilhac-Benedetti, following ideas of Kashaev, introduced invariants of pseudo-Anosov homeomorphisms of punctured hyperbolic surfaces that depend on a complex root of unity of odd order. Around the same time, Bonahon-Liu introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. A little later, Dimofte and the first author introduced invariants of cusped hyperbolic 3-manifolds at roots of unity using their geometric representation. In another effort, Bonahon-Wong-Yang introduced another set of invariants of pseudo-Anosov homeomorphisms at roots of unity. All these invariants are conjecturally closely related, and our aim is to prove a precise relation between the Baseilhac-Benedetti invariants, the Bonahon-Liu-Wong-Yang and the lesser-known abelian $\mathfrak{gl}_1$-invariants.

math.GT↗

arXiv2025

On the colored Links--Gould polynomial

We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the cabling formula, we obtain genus bounds and a specialization to the Alexander polynomial for the colored Links--Gould polynomial that is independent of $n$, which implies corresponding properties of the $V_n$-polynomial for $n=2$ conjectured in previous work of two of the authors, and extends the work done for $n=1$. Combined with work of one of the authors arXiv:2409.03557, our genus bound for $\mathrm{LG}^{(2)}=V_2$ is sharp for all knots with up to $16$ crossings.

math.QA↗

arXiv2025

On the Bonahon--Wong--Yang invariants of pseudo-Anosov maps

We conjecture (and prove for once-punctured torus bundles) that the Bonahon--Wong--Yang invariants of pseudo-Anosov homeomorphisms of a punctured surface at roots of unity coincide with the 1-loop invariant of their mapping torus at roots of unity. This explains the topological invariance of the BWY invariants and how their volume conjecture, to all orders, and with exponentially small terms included, follows from the quantum modularity conjecture. Using the numerical methods of Zagier and the first author, we illustrate how to efficiently compute the invariants and their asymptotics to arbitrary order in perturbation theory, using as examples the $LR$ and the $LLR$ pseudo-Anosov monodromies of the once-punctured torus. Finally, we introduce descendant versions of the 1-loop and BWY invariants and conjecture (and numerically check for pseudo-Anosov monodromies of $L/R$-length at most 5) that they are related by a Fourier transform. This edition includes statements and proofs for roots of unity of all order, even and odd.

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arXiv2025

The Habiro ring of a number field

We introduce the Habiro ring of a number field $\mathbb{K}$ and modules over it graded by $K_3(\mathbb{K})$. Elements of these modules are collections of power series at each complex root of unity that arithmetically glue with each other after applying a Frobenius endomorphism, and after dividing at each prime by a collection of series that depends solely on an element of the Bloch group. The main theorems of this paper concern number fields, their algebraic $K$-theory and its regulator maps (Borel, $p$-adic and étale), whereas the explicit collections of series are defined by a careful algebraic analysis of the infinite Pochhammer symbol at roots of unity. The origin of the above mentioned power series comes from perturbative Chern--Simons theory and by expansions of the admissible series of Kontsevich--Soibelman, both ultimately related to the infinite Pochhammer symbol. This link suggests that some Donaldson-Thomas invariants have arithmetic meaning and that some elements of the Habiro ring of a number field have enumerative meaning. Added subsection 1.1 explaining what the paper is about and subsection 1.8 explaining the relation to perturbative complex Chern-Simons theory.

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arXiv2025

Explicit classes in Habiro cohomology

We propose a cycle description of the Habiro cohomology of a smooth variety $X$ over the spectrum $B$ of an étale $Z[λ]$-algebra and construct explicit nontrivial cycles using either the Picard-Fuchs equation on $X/B$ of a hypergeometric motive, or a push-forward of elements of the Habiro ring of $X/B$. In particular, we give explicit classes for 1-parameter Calabi--Yau families. The $q$-hypergeometric origin of our cycles imply that they generate $q$-holonomic modules that define $q$-deformations of the classical Picard-Fuchs equation. We illustrate our theorems with three examples: the Legendre family of elliptic curves, the $A$-polynomial curve of the figure eight knot, and for the quintic three-fold, whose $q$-Picard Fuchs equation appeared in its genus $0$-quantum $K$-theory. Our methods give a unified treatment of quantum $K$-theory and complex Chern-Simons theory around higher dimensional critical loci.

math.AG↗

arXiv2024

1-loop equals torsion for fibered 3-manifolds

In earlier work of two of the authors, two 1-loop polynomial invariants of cusped 3-manifolds were constructed using combinatorial data of ideal triangulations, and conjectured to be equal to the $\mathbb{C}^2$ and the $\mathbb{C}^3$-torsion polynomials. Here, we prove this conjecture for layered triangulations of fibered 3-manifolds with toroidal boundary, and we illustrate our theorems with exact computations of the 1-loop and the torsion polynomials. As further evidence for the conjecture, we confirm it for more than 6,600 nonfibered manifolds, and use this data to explore the extent to which the $\mathbb{C}^2$-torsion determines the Thurston norm.

math.GT↗

arXiv2024

1-loop equals torsion for two-bridge knots

Motivated by the conjectured asymptotics of the Kashaev invariant, Dimofte and the first author introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic 3-manifold, proved that its constant (1-loop) term is a topological invariant and conjectured that it equals to the adjoint Reidemeister torsion. We prove this conjecture for hyperbolic 2-bridge knots by combining the work of Ohtsuki--Takata with an explicit computation.

math.GT↗

arXiv2024

Algebraic aspects of holomorphic quantum modular forms

Matrix-valued holomorphic quantum modular forms are intricate objects that arise in successive refinements of the Volume Conjecture of knots and involve three holomorphic, asymptotic and arithmetic objects. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the $(-2,3,7)$-pretzel knot.

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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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