arXiv subjects

Javier Martínez-Aguinaga

Publications and source records attributed to Javier Martínez-Aguinaga.

12 recordsLinked to original sources

arXiv2026

Families of knots that cannot be made Legendrian parametrically

The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\geq 3$, every knot type $\mathcal K$, every Legendrian representative $\mathcal L$ and every formal Legendrian representative $\mathcal{FL}$, the associated group homomorphisms $π_n(\mathcal{L})\toπ_n(\mathcal{K})$ and $π_n(\mathcal{FL})\toπ_n(\mathcal{K})$ are never surjective. We then show that surjectivity at the $π_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $π_1$.

math.GT↗

arXiv2026

Spaces of tight contact structures in dimension 3

We study spaces of tight contact structures and contactomorphism groups in dimension $3$. We introduce a microfibration method for studying spaces of convex surfaces in contact $3$-manifolds and apply it to describe spaces of convex disks and spheres in tight contact $3$-manifolds. With these ingredients in hand, we prove a parametric gluing theorem for contact handle attachments: the weak homotopy type of the space of tight contact structures is invariant under contact $0$-, $1$-, and $3$-handle attachments. Based on these new techniques, we present a substantial number of independent applications. First, we obtain explicit descriptions of the weak homotopy type of several spaces of tight contact structures. We determine the weak homotopy type of the contactomorphism group of every standard contact handlebody, of the standard $\mathbb{S}^1\times\mathbb{S}^2$, and of every Legendrian circle bundle with non-empty boundary, partially resolving a conjecture of Giroux. Additionally, we compute the weak homotopy type of an infinite family of spaces of Legendrian and transverse unknots, proving Legendrian and transverse analogues of Hatcher's theorem for smooth unknots in the $3$-sphere.

math.SG↗

arXiv2026

On the Legendrian realisation of loops of knots

We study the natural inclusion of the space of Legendrian embeddings in $(\mathbb{S}^3,ξ_{\operatorname{std}})$ into the space of smooth embeddings at the level of fundamental groups. T. Kálmán posed in [Kal] the open question of whether for every fixed knot type $\mathcal{K}$ and Legendrian representative $\mathcal{L}$, the homomorphism $π_1(\mathcal{L})\toπ_1(\mathcal{K})$ is surjective. We positively answer this question for infinitely many knot types $\mathcal{K}$ in the three main families (hyperbolic, torus and satellites) and every stabilised Legendrian representative in $(\mathbb{S}^3,ξ_{\operatorname{std}})$.

math.GT↗

arXiv2026

Fat distributions with Reeb directions need not be complex contact

It is well known that every complex contact $3$-manifold, when regarded as a real manifold, gives rise to a fat $(4,6)$-distribution that admits two Reeb directions. Nonetheless, it was an open question whether the converse was true. This was not known even at the level of germs. The present work completely answers this question in the negative. We construct the first example of a fat distribution with two Reeb directions that does not support a complex contact structure anywhere, not even locally nor up to diffeomorphism. This result answers an open question by Aritra Bhowmick.

math.GT↗

arXiv2026

A constructive solution to the equivalence problem for knot projectivizations

The problem of whether different projectivizations of the same affine knot $K\subset\mathbb{S}^3$ are equivalent in $\mathbb{R}\mathbb{P}^3$ can be found in [11] and has also been posed as an open question in [15]. In this note we provide a constructive solution to the problem. In particular, we adapt an idea due to A. Hatcher developed in the realm of embedding spaces and we describe an algorithm that produces an explicit isotopy between any two given projectivizations of the same affine knot. More generally, we introduce the notion of lensification of a knot in any lens space $L(p,q)$ and describe an algorithm that works in that more general setting, of which $\mathbb{R}\mathbb{P}^3\simeq L(2,1)$ is a particular instance. Finally, we apply this algorithm to several pairs of knots from the literature for which the equivalence problem was raised as an open question, finding explicit isotopies.

math.GT↗

arXiv2024

On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

We study the global topology of the space $\mathcal L$ of loops of contactomorphisms of a non-orderable closed contact manifold $(M^{2n+1}, α)$. We filter $\mathcal L$ by a quantitative measure of the ``positivity'' of the loops and describe the topology of $\mathcal L$ in terms of the subspaces of the filtration. In particular, we show that the homotopy groups of $\mathcal L$ are subgroups of the homotopy groups of the subspace of positive loops $\mathcal L^+$. We obtain analogous results for the space of loops of Legendrian submanifolds in $(M^{2n+1}, α)$.

math.SG↗

arXiv2024

Parametric satellites and connected-sums in the space of Legendrian embeddings

This article introduces two new constructions at the higher homotopy level in the space of Legendrian embeddings in $(\mathbb{R}^3, ξ_{\operatorname{std}})$. We first introduce the parametric Legendrian satellite construction, showing that the satellite operation works for parametric families of Legendrian embeddings. This yields new invariants at the higher-order homotopy level. We then introduce the parametric connected-sum construction. This operation takes as inputs two $n$-spheres based at Legendrian embeddings $K_1$ and $K_2$, respectively, and produces a new $n$-sphere based at $K_1\# K_2$. As a main application we construct new infinite families of loops of Legendrian embeddings with non-trivial LCH monodromy invariant.

math.SG↗

arXiv2023

Existence and classification of maximal growth distributions

This article tackles the problem of existence and classification of maximal growth distributions on smooth manifolds. We show that maximal growth distributions of rank$>2$ abide by a full $h$-principle in all dimensions. We make use of M. Gromov's higher order convex integration and, on the way, we establish a new criterion for checking ampleness of a differential relation. As a consequence we answer in the positive, for $k>2$, the long-standing open question posed by M. Kazarian and B. Shapiro more than 25 years ago in [14] of whether any parallelizable manifold admits a $k$-rank distribution of maximal growth. We also answer several related open questions. For completeness we show that the differential relation of maximal growth for rank-$2$ distributions is not ample in any ambient dimension. Non-ampleness of the Engel and the $(2,3,5)$-conditions follow as particular cases.

math.GT↗

arXiv2022

Classification of tangent and transverse knots in bracket-generating distributions

Consider a manifold, of dimension greater than 3, equipped with a bracket-generating distribution. In this article we prove complete h-principles for embedded regular horizontal curves and for embedded transverse curves. These results contrast with the 3-dimensional contact case, where the full h-principle for transverse/legendrian knots is known not to hold. We also prove analogous statements for immersions, with no assumptions on the ambient dimension.

math.DG↗

arXiv2021

Convex integration with avoidance and hyperbolic (4,6) distributions

This paper tackles the classification, up to homotopy, of tangent distributions satisfying various non-involutivity conditions. All of our results build on Gromov's convex integration. For completeness, we first prove that that the full h-principle holds for step-2 bracket-generating distributions. This follows from classic convex integration, no refinements of the theory are needed. The classification of (3,5) and (3,6) distributions follows as a particular case. We then move on to our main example: A complete h-principle for hyperbolic (4,6) distributions. Even though the associated differential relation fails to be ample along some principal subspaces, we implement an "avoidance trick" to ensure that these are avoided during convex integration. Using this trick we provide the first example of a differential relation that is ample in coordinate directions but not in all directions, answering a question of Eliashberg and Mishachev. This so-called "avoidance trick" is part of a general avoidance framework, which is the main contribution of this article. Given any differential relation, the framework attempts to produce an associated object called an "avoidance template". If this process is successful, we say that the relation is "ample up to avoidance" and we prove that convex integration applies. The example of hyperbolic (4,6) distributions shows that our framework is capable of addressing differential relations beyond the applicability of classic convex integration.

math.DG↗

arXiv2020

Fundamental groups of formal Legendrian and horizontal embedding spaces

We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in R3. We use it to show that previous examples in the literature of non trivial loops of Legendrian embeddings are already non trivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel R4. We check that this computes the fundamental group of the space of horizontal embeddings as well.

math.SG↗

arXiv2018

Loops of Legendrians in contact 3-manifolds

We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.

math.SG↗