arXiv subjects

José Pedro Quintanilha

Publications and source records attributed to José Pedro Quintanilha.

12 recordsLinked to original sources

arXiv2026

All knots are trivial: a "proof" by sleight of hand

We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a trick by depicting the target knot as a special type of knot diagram, which we call a "knotholder diagram". By proving that all knots admit knotholder diagrams, we obtain variants of the trick for producing every knot.

math.GT↗

arXiv2026

Geometric invariants of locally compact groups: the homotopical perspective

We extend the classical theory of homotopical $Σ$-sets $Σ^n$ developed by Bieri, Neumann, Renz and Strebel for abstract groups, to $Σ$-sets $Σ_{\mathrm{top}}^n$ for locally compact Hausdorff groups. Given such a group $G$, our $Σ_{\mathrm{top}}^n(G)$ are sets of continuous homomorphisms $G \to \mathbb{R}$ ("characters"). They match the classical $Σ$-sets $Σ^n(G)$ if $G$ is discrete, and refine the homotopical compactness properties $\mathrm C_n$ of Abels and Tiemeyer. Moreover, our theory recovers the definition of $Σ_{\mathrm{top}}^1$ and $Σ_{\mathrm{top}}^2$ proposed by Kochloukova. Besides presenting various characterizations of $Σ_{\mathrm{top}}^n$ (particularly for $n\in \{1,2\}$), we show that characters in $Σ_{\mathrm{top}}^n(G)$ are also in $Σ_{\mathrm{top}}^n(H)$ if $H\le G$ is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of $Σ_{\mathrm{top}}^n(G)$ is open, we prove that characters in a group of type $\mathrm C_n$ that do not vanish on the center always lie in $Σ_{\mathrm{top}}^n(G)$, and we relate the $Σ$-sets of a group with those of its quotients by closed subgroups of type $\mathrm C_n$. Lastly, we describe how $Σ_{\mathrm{top}}^n(G)$ governs whether a closed normal subgroup with abelian quotient is of type $\mathrm C_n$, generalizing one of the highlights of the classical theory.

math.GR↗

arXiv2026

Genus-zero links with prescribed knots as components

We prove that any finite collection of at least three isotopy classes of knots in a 3-manifold $M$ is realizable as the components of a genus-zero link in $M$, provided that an obvious requirement on their conjugacy classes in $π_1(M)$ is met. This condition is vacuously satisfied for $M = \mathbb S^3$, and in this case we also control the pairwise linking numbers of the components. Replacing the 3-genus with the 4-genus, we obtain an analogous result where only two knot isotopy classes are prescribed.

math.GT↗

arXiv2026

Cobordism-equivalence for codimension-one submanifolds

We show that two hypersurfaces in a manifold are related by a sequence of embedded cobordisms if and only if they represent the same homology class. By applying handle decompositions we turn these cobordisms into a sequence of embedded surgeries. Specializing to Seifert surfaces we obtain a conceptual proof that two Seifert surfaces of a fixed link are related by tube attachments and tube removals.

math.GT↗

arXiv2025

Algorithms in 4-manifold topology

We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric. In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic.

math.GT↗

arXiv2025

Geometric invariants of locally compact groups: the homological perspective

In this paper we develop the homological version of $Σ$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $Σ$-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type $\mathrm{CP}_m$ and homological locally compact $Σ^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $Σ^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $Σ$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$.

math.AT↗

arXiv2025

Geometric invariants of TDLC completions

Recently, Bonn and Sauer showed that, from the point of view of compactness properties, the Schlichting completion of a Hecke pair $(Γ,Λ)$ behaves precisely as if it were the quotient of $Γ$ by $Λ$. Motivated by this result, we prove that a similar phenomenon holds for the $Σ$-sets. More generally, we relate the $Σ$-sets of every TDLC completion of a Hecke pair $(Γ,Λ)$ to the $Σ$-sets of $Γ$ whenever $Λ$ satisfies suitable compactness properties. We provide applications to TDLC completions of Baumslag-Solitar groups and certain groups of upper triangular matrices studied by Schesler.

math.GR↗

arXiv2024

Writhe invariants of 3-regular spatial graphs

We give a necessary condition for two diagrams of $3$-regular spatial graphs with the same underlying abstract graph $G$ to represent isotopic spatial graphs. The test works by reading off the writhes of the knot diagrams coming from a collection of cycles in $G$ in each diagram, and checking whether the writhe tuples differ by an element in the image of a certain map of $\mathbb{Z}$-modules determined by $G$. We exemplify by using our result to distinguish, for each $n \ge 3$, all elements in a certain infinite family of embeddings of the Möbius ladder $\mathrm{M}_n$ into $\mathbb{R}^3$ . We also connect these writhe tuples to a classical invariant of spatial graphs due to Wu and Taniyama.

math.GT↗

arXiv2022

An algorithm to calculate generalized Seifert matrices

We develop an algorithm for computing generalized Seifert matrices for colored links given as closures of colored braids. The algorithm has been implemented by the second author as a computer program called Clasper. Clasper also outputs the Conway potential function, the multivariable Alexander polynomial and the Cimasoni-Florens signatures of a link, and displays a visualization of the C-complex used for producing the generalized Seifert matrices.

math.GT↗

arXiv2022

The Complex of Hypersurfaces in a Homology Class

For a compact oriented smooth $n$-manifold $M$ and a codimension-$1$ homology class $ϕ\in \operatorname{H}_{n-1}(M, \partial M)$, we investigate a simplicial complex $\mathcal{S}^\dagger(M, ϕ)$ relating the properly embedded hypersurfaces in $M$ representing $ϕ$. Its definition is akin to that of other classical complexes, such as the curve complex of a surface or the Kakimizu complex of a knot, with the difference that hypersurfaces are not taken up to isotopy. We prove that $\mathcal{S}^\dagger(M, ϕ)$ is connected and simply connected in every dimension $n$. We also show connectedness of a similar complex $\mathcal{T}^\dagger(M, ϕ)$ adapted to the $3$-dimensional case, where only Thurston norm-realizing surfaces are considered. The connectedness results are transported to the complexes $\mathcal{S}(M, ϕ), \mathcal{T}(M, ϕ)$ where hypersurfaces are taken up to isotopy, and for $n=2$ the simple connectedness result carries over as well. We also briefly discuss extensions to a context studied by Turaev, where regular graphs in $2$-complexes are used to represent $1$-dimensional cohomology classes. We finish with two applications: we give an alternative proof of the fact that all Seifert surfaces for a fixed knot in a rational homology sphere are tube-equivalent, and we use connectedness of $\mathcal{T}^\dagger(M, ϕ)$ to define a new $\ell^2$-invariant of $2$-dimensional homology classes in irreducible and boundary-irreducible oriented compact connected $3$-manifolds with empty or toroidal boundary.

math.GT↗

arXiv2021

Canonical decompositions and algorithmic recognition of spatial graphs

We prove that there exists an algorithm for determining whether two piecewise-linear spatial graphs are isomorphic. In its most general form, our theorem applies to spatial graphs furnished with vertex colorings, edge colorings and/or edge orientations. We first show that spatial graphs admit canonical decompositions into blocks, that is, spatial graphs that are non-separable and have no cut vertices, in a suitable topological sense. Then we apply a result of Haken and Matveev in order to algorithmically distinguish these blocks.

math.GT↗

arXiv2021

Stable integral simplicial volume of 3-manifolds

We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.

math.GT↗