arXiv subjects

Mario Eudave-Muñoz

Publications and source records attributed to Mario Eudave-Muñoz.

9 recordsLinked to original sources

arXiv2026

A Kuratowski-Type Classification of Critical Complexes for the 3-Sphere

We give a Kuratowski-type classification of a graph-defined class of minimal piecewise-linear obstructions to embeddability in the 3-sphere. A finite simplicial complex $X$ is called critical for $S^3$ if $|X|$ does not embed in $S^3$, whereas deleting the open star of any simplex in the second barycentric subdivision of $X$ yields a polyhedron embeddable in $S^3$. The main theorem completely classifies critical complexes of the form $(G\times S^1)\cup H$, where $G$ and $H$ are graphs and $H$ is attached along vertices of the branch set of $G\times S^1$. We prove that there are exactly seven such complexes up to homeomorphism: two $K_4$-type complexes, four $Θ_4$-type complexes, and one $K_{2,3}$-type complex. The proof is combinatorial in nature. By collapsing the $S^1$-factor of $G\times S^1$, we associate to $X$ a reduction graph $\widehat X=G\cup H$. Criticality implies that $H$ is a forest, $G$ is planar, and $\widehat X$ is inclusion-minimal non-planar. Kuratowski's theorem therefore reduces the classification to the cases $K_5$ and $K_{3,3}$. A finite analysis of forest attachments, together with a face-incidence criterion for embeddability, leaves precisely the seven models listed above. We also prove that every non-embeddable regular multibranched surface in $S^3$ contains a critical subcomplex of the form $M\cup H$, where $M$ is a regular multibranched surface and $H$ is a graph.

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arXiv2024

Positive Artin Presentations

An integral framed, closed pure n-braid B' in the 3-sphere describes a positive Artin presentation, if the braid B can be put on a disk with holes such that each relation describes a positive path and these paths are disjoint. In the present paper we classify the closed, pure n-braids B' in the 3-sphere, such that B represents a positive Artin presentation. Also we prove that if such B describes a positive Artin presentation then B' is strongly invertible. Such positive Artin presentation give us the fundamental group of closed, connected and orientable 3-manifolds M, and in fact, by giving an example, we show that there exist 3-manifolds whose fundamental group does not admit a positive Artin presentation.

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arXiv2023

Hyperbolic knots with a large toroidal surgery

We show an infinite family of hyperbolic knots that have an exceptional surgery producing a graph manifold containing five disjoint, and non parallel incompressible tori.

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arXiv2023

On the transient number of a knot

The transient number of a knot K, denoted tr(K), is the minimal number of simple arcs that have to be attached to K, in order that K can be homotoped to a trivial knot in a regular neighborhood of the union of K and the arcs. We give a lower bound for tr(K) in terms of the rank of the first homology group of the double branched cover of K. In particular, if t(K)=1, then the first homology group of the double branched cover of K is cyclic. Using this, we can calculate the transient number of many knots in the tables and show that there are knots with arbitrarily large transient number.

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arXiv2021

On non almost-fibered knots

An almost-fibered knot is a knot whose complement possesses a circular thin position in which there is one and only one weakly incompressible Seifert surface and one incompressible Seifert surface. Infinite examples of almost-fibered knots are known. In this article, we show the existence of infinitely many hyperbolic genus one knots that are not almost-fibered.

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arXiv2020

Satellite knots and trivializing bands

We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.

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arXiv2019

Computing Genera of Satellite Tunnel Number One Knots and Torti-rational Knots

The genus of satellite tunnel number one knots and torti-rational knots is computed using the tools introduced by Floyd and Hatcher. An implementation of an algorithm is given to compute genus and slopes of minimal genus Seifert surfaces for such knots.

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arXiv2016

Classification of genus two knots which admit a (1,1) decomposition

We describe the genus two knots which admit a genus one, one bridge position. These are divided into several families, one consists of vertical bandings of two genus one $(1,1)$-knots, other consists of vertical bandings of two cross cap number two 2-bridge knots, and the last one consists of genus two tunnel number one satellite knots.

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arXiv1997

Nonhyperbolic Dehn fillings on hyperbolic 3-manifolds

We give three infinite families of examples of nonhyperbolic Dehn fillings on hyperbolic manifolds. A manifold in the first family admits two Dehn fillings of distance two apart, one of which is toroidal and annular, and the other is reducible and $\partial$-reducible. A manifold in the second family has boundary consisting of two tori, and admits two reducible Dehn fillings. A manifold in the third family admits a toroidal filling and a reducible filling with distance 3 apart. These examples establish the virtual bounds for distances between certain types of nonhyperbolic Dehn fillings.

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