arXiv subjects

José Andrés Rodríguez Migueles

Publications and source records attributed to José Andrés Rodríguez Migueles.

11 recordsLinked to original sources

arXiv2026

The Lorenz braid index and hyperbolic volume

A result of Futer, Kalfagianni, and Purcell implies that an upper volume bound for all link complements in the 3-sphere cannot depend solely on the braid index. In this paper, we introduce the Lorenz braid index and generalise the bunch algorithm to provide a general upper volume bound for all link complements in the 3-sphere. Such an upper bound is a quadratic polynomial in the Lorenz braid index. In addition, we construct an explicit family of hyperbolic Lorenz knots for which the classical braid index and the Seifert genus both tend to infinity, while the Lorenz braid index remains bounded.

math.GT↗

arXiv2026

Generalised T-links Represent All Links in the 3-Sphere

We introduce generalised T-links, a restricted extension of the widely studied families of T-links and twisted torus links, and prove that this family is universal: every link in the $3$-sphere can be represented as a generalised T-link. This gives a compact braid-theoretic form of Ghrist's universality theorem for certain Lorenz-like templates. More precisely, we replace the template-theoretic representatives coming from universal Lorenz-like templates with explicit braid closures depending on finitely many integer parameters. Although closely related braid families have appeared previously in the literature, the universality of this restricted class is the main point of the present paper. Thus, all links in the $3$-sphere are placed inside a natural structured class closely related to T-links, Lorenz links, and twisted torus links.

math.GT↗

arXiv2025

On the existence of universal links in three-manifolds

We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish between universal links and complement universal links and show that these notions do not coincide in general, by exhibiting infinitely many examples of complement universal links that are not universal. Also, we prove that there is no closed aspherical $3$-manifold, such that every closed, aspherical $3$-manifold is a branched covering over it. Finally, we characterize the closed $3$-manifolds admitting branching coverings from $P^3 \# P^3$, and deduce that there is no closed reducible $3$-manifold, such that every closed reducible $3$-manifold is a branched covering over it.

math.GT↗

arXiv2025

Modular links: Bunch algorithm and upper volume bounds

In the 1970s, Williams developed an algorithm that has been used to construct and study modular links in the Lorenz template. We introduce an improved algorithm, which we call the bunch algorithm, to provide more insights into the geometry of modular links and Lorenz links. Using the machinery developed for the bunch algorithm, we provide the first upper volume bound that is independent of word exponents and quadratic in the braid index of the Lorenz link component for all modular link complements. We find families of modular knot complements with upper volume bounds that are linear in the braid index. A classification of modular link complements based on the relative magnitudes of word exponents is also presented.

math.GT↗

arXiv2024

Geometry and topology of closed geodesics complements in the 3-torus

We show that for at most three closed geodesics with linearly independent directions, the homeomorphism type of its complement in the 3-torus is determine by the orbit of their direction vectors subspaces under the action of $PSL_3(\mathbb{Z}).$ Moreover, we provide asymptotically sharp volume bounds for a family of closed geodesics complements. The bounds depend only on the distance in the Farey graph.

math.GT↗

arXiv2024

A lower bound for the volumes of modular link complements

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link complement with respect to its unique complete hyperbolic metric. We provide the first lower volume bound that is linear in terms of the number of distinct exponents in the code words corresponding to the collection of closed geodesics.

math.GT↗

arXiv2024

Arithmetic modular links

We construct a sequence of geodesics on the modular surface such that the complement of the canonical lifts to the unit tangent bundle are arithmetic 3-manifolds.

math.GT↗

arXiv2023

Periods of continued fractions and volumes of modular knots complements

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic's continued fraction expansion. Consequently, we give a volume's upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index. Also, for any punctured hyperbolic surface we give volume's bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length.

math.GT↗

arXiv2022

On volumes and filling collections of multicurves

Let $S$ be a surface of negative Euler characteristic and consider a finite filling collection $Γ$ of closed curves on $S$ in minimal position. An observation of Foulon and Hasselblatt shows that $PT(S) \setminus \hatΓ$ is a finite-volume hyperbolic 3-manifold, where $PT(S)$ is the projectivized tangent bundle and $\hatΓ$ is the set of tangent lines to $Γ$. In particular, $vol(PT(S) \setminus \hatΓ)$ is a mapping class group invariant of the collection $Γ$. When $Γ$ is a filling pair of simple closed curves, we show that this volume is coarsely comparable to Weil-Petersson distance between strata in Teichmüller space. Our main tool is the study of stratified hyperbolic links $\barΓ$ in a Seifert-fibered space $N$ over $S$. For such links, the volume of $N\setminus\barΓ$ is coarsely comparable to expressions involving distances in the pants graph.

math.GT↗

arXiv2020

A lower bound for the volumes of complements of periodic geodesics

Every closed geodesic $γ$ on a surface has a canonically associated knot $\widehatγ$ in the projective unit tangent bundle. We study, for $γ$ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of homotopy classes of $γ$-arcs in each pair of pants of a pants decomposition of the surface.

math.GT↗

arXiv2019

Hyperbolicity of links complements in Seifert fibered spaces

Let $\barγ$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $γ$ in $\mathcal O.$ We prove that the complement $M_{\barγ}$ of $\barγ$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.

math.GT↗