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Thiago de Paiva

Publications and source records attributed to Thiago de Paiva.

At least 19 recordsLinked to original sources
arXiv2026

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

Generalised $T$-links provide a simple description of all links in $S^3$ as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

math.GT↗
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

Lorenz Links that are not Horseshoe and Rossler Links

We compare the periodic orbit types of the Lorenz, horseshoe, and Rössler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the Rössler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this paper, Rössler links and horseshoe links are treated as belonging to the same template class. We show that the relationship between Lorenz links and horseshoe/Rössler links has two complementary sides. First, we prove that the overlap between the two families is nontrivial by constructing infinite families of links which can be embedded in both templates. This verifies, for these families, a conjecture stated by Kofman that horseshoe links should also be Lorenz links. On the other hand, we prove that the two families are far from being the same. Holmes and Williams showed that many Lorenz torus knots cannot be embedded in the horseshoe template: if the torus knot \(T(p,q)\), with \(p<q\), is a horseshoe knot, then \(3p\leq 2q\). We show that this phenomenon is much broader. We extend the Holmes--Williams obstruction from torus knots to torus links, and we construct infinitely many hyperbolic Lorenz knots and links, as well as infinitely many satellite Lorenz knots and links, which cannot be embedded in the horseshoe template. Consequently, these examples are Lorenz links which are not horseshoe links and hence not Rössler links.

math.GT↗
arXiv2026

Essential tori associated with links of mixed singularities

We establish a direct connection between the analytic data of weakly isolated mixed singularities and the topology of their associated links. More precisely, we prove that the existence of essential tori, topological information, in the complements of links arising from weakly isolated mixed singularities can be detected directly from properties of the defining mixed polynomial, provided that it is convenient, non-degenerate and $Γ$-nice. Our results provide explicit and computable criteria, expressed purely in terms of the polynomial data, that determine the presence of essential tori in the link exterior. In particular, these criteria yield effective conditions ensuring that such links are non-hyperbolic. This approach provides a new method to extract topological information about link complements without requiring an explicit determination of the link type, thereby establishing a concrete bridge between the analytic structure of mixed polynomials and the geometric topology of their associated links.

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↗
arXiv2026

Satellite Links with Multiple Full Twists and Single-Twist Companions

We study the relationship between the number of full twists in positive braid representations of satellite links and their companion links. We construct infinitely many satellite links that admit positive braid representations with arbitrarily many full twists, while their companion links do not admit any positive braid representation with more than one full twist. This exhibits an unexpected divergence between the braid-theoretic complexity of a satellite link and that of its companion.

math.GT↗
arXiv2026

Lorenz links, T-links, Minimal Braids, Positive braids with a full twist, and Geometric Types

Lorenz links and T-links are equivalent families by the work of Birman--Kofman. Lorenz links arise as periodic orbits of the Lorenz system, whereas T-links are closures of certain positive braids. Birman, Williams, and Franks showed that every Lorenz link admits a positive braid with a full twist realizing the braid index. We study Lorenz links from the minimal-braid viewpoint. We compute these representatives from the T-link parameters and show that they are precisely a family of positive braids with a full twist, called V-link braids. This gives an explicit correspondence between V-links and T-links. We use this correspondence to study equivalences among T-link presentations. We obtain criteria for distinguishing T-links from their parameters, recover the Birman--Kofman equivalence from the V-link/T-link correspondence, and prove that, under mild assumptions, this equivalence gives two distinct presentations of the same link. We also generalize this phenomenon: under natural non-degeneracy conditions, one obtains four distinct T-link presentations of the same link. More generally, even with the braid index fixed, the number of distinct V-link and T-link presentations can be arbitrarily large. In contrast, we initiate the study of uniqueness of T-link presentations. We prove that there are infinite families of Lorenz links with a unique T-link presentation, after excluding the destabilization case. For the Lorenz flow, this gives families of periodic orbits that admit only one T-link parameter description in their isotopy class. Finally, the V-link form places Lorenz links in the setting of positive braids with a full twist, where techniques for detecting geometric type apply. We prove criteria for such braids to be satellite or hyperbolic and apply them to V-links and T-links. As a consequence, we obtain new families of satellite and hyperbolic T-links and generalize known results.

math.GT↗
arXiv2026

Hyperbolic Twisted Torus Links

The twisted torus link $T(p, q; r, s)$ is obtained by twisting $r$ parallel strands of the $(p, q)$-torus link a total of $s$ full times. In this paper we find all twisted torus links which are hyperbolic for $\vert s\vert >3$.

math.GT↗
arXiv2026

Twisted torus links that are unlinks

A twisted torus link $T(p,q,r,s)$ is obtained by performing $s$ full twists on $r$ adjacent strands of the $(p,q)$-torus link. In this paper, we classify twisted torus links that are unlinks. We give a complete characterization of all parameter families $(p,q,r,s)$ for which the associated twisted torus link is an unlink.

math.GT↗
arXiv2025

On the number of components of twisted torus links

Twisted torus links $T(p,q;r,s)$ generalize torus links by introducing $s$ additional twists on $r$ adjacent strands of the torus link $T(p,q)$. It is well known that the number of components of a torus link $T(p, q)$ is given by the greatest common divisor of $p$ and $q$. However, determining the number of components of twisted torus links is not as straightforward based solely on their parameters. In this work, we present a Euclidean algorithm-like procedure for computing the number of components of twisted torus links based on their parameters. As a result, we show that the number of components of a twisted torus link $T(p, q; r, s)$ is a multiple of $\gcd(p, q, r, s)$, and in particular, $T(p, q; r, s)$ is a knot only if $\gcd(p, q, r, s) = 1$. We also use our algorithm to prove several conjectures related to the number of components in twisted torus links.

math.GT↗
arXiv2024

Crushing Surfaces of Positive Genus

The operation of crushing a normal surface has proven to be a powerful tool in computational $3$-manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can drastically change the topology of a triangulation, so applications to date have been limited to relatively simple surfaces: $2$-spheres, discs, annuli, and closed boundary-parallel surfaces. We give the first detailed analysis of the topological effects of crushing closed essential surfaces of positive genus. To showcase the utility of this new analysis, we use it to prove some results about how triangulation complexity interacts with JSJ decompositions and satellite knots; although similar applications can also be obtained using techniques of Matveev, our approach has the advantage that it avoids the machinery of almost simple spines and handle decompositions.

math.GT↗
arXiv2024

Hyperbolic and satellite Lorenz links obtained by twisting

A Lorenz link is equivalent to a T-link, which is a positive braid built by concatenating torus braids of increasing size. When each torus braid except the largest is obtained by full twists, then the T-link can be described as the Dehn filling of a parent link. In this paper, we completely classify when such parent links are hyperbolic. This gives a classification of the geometry of T-links obtained by full twists when the amount of twisting is large, although the bound on the number of required twists is not effective. We also present effective results on hyperbolicity for two families of T-links obtained by twisting, even when the number of twists is small. Finally, we identify families of satellite T-links obtained by half-twists.

math.GT↗
arXiv2024

Geometry of Generalized Augmented Braids

A generalized augmented link of a knot $K$ is a link obtained by adding trivial components to $K$ that bound $n$-punctured disks. In this paper we consider that $K$ is given by a positive braid with at least one full twist. We characterize when the generalized augmented links of $K$ have hyperbolic geometry.

math.GT↗
arXiv2023

Satellite knots that cannot be represented by positive braids with full twists

A positive braid with at least one full twist is known to be a minimal braid, i.e, it achieves the braid index for its closure. In this paper we find knots that are the closure of positive minimal braids that cannot be represented by positive braids with full twists. More precisely, we show that some satellite knots with companions and patterns given as the closure of positive braids cannot be represented as the closure of positive braids with full twists. As a consequence, we find infinitely many satellite knots with companions and patterns being Lorenz knots that are not Lorenz knots. This gives an answer to the question whether a satellite knot having Lorenz pattern and companion is also a Lorenz knot, originally addressed by Birman and Williams in a special case in the 1980s.

math.GT↗
arXiv2023

Torus Lorenz Links obtained by Full Twists along Torus Links

All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman.

math.GT↗
arXiv2022

Satellites and Lorenz knots

We construct infinitely many families of Lorenz knots that are satellites but not cables, giving counterexamples to a conjecture attributed to Morton. We amend the conjecture to state that Lorenz knots that are satellite have companion a Lorenz knot, and pattern equivalent to a Lorenz knot. We show this amended conjecture holds very broadly: it is true for all Lorenz knots obtained by high Dehn filling on a parent link, and other examples.

math.GT↗
arXiv2021

Unexpected Essential Surfaces among Exteriors of Twisted Torus Knots

The twisted torus knots K(p, q; r, s) are obtained by performing a sequence of s full twists on r adjacent strands of (p, q)-torus knots. Morimoto asked whether all twisted torus knots with essential tori in the exterior fit into one of two families. We prove that the answer to this question is no, by finding two different new families of toroidal twisted torus knots.

math.GT↗