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Mario Eudave-Munoz

Publications and source records attributed to Mario Eudave-Munoz.

13 recordsLinked to original sources
arXiv2020

The maximum and minimum genus of a multibranched surface

In this paper, we give a lower bound for the maximum and minimum genus of a multibranched surface by the first Betti number and the minimum and maximum genus of the boundary of the neighborhood of it, respectively. As its application, we show that the maximum and minimum genus of $G\times S^1$ is equal to twice of the maximum and minimum genus of $G$ for a graph $G$, respectively. This provides an interplay between graph theory and 3-manifold theory.

math.GT↗
arXiv2018

Characterization of 3-punctured spheres in non-hyperbolic link exteriors

In this paper, we characterize non-hyperbolic 3-component links in the 3-sphere whose exteriors contain essential 3-punctured spheres with non-integral boundary slopes. We also show the existence of embeddings of some multibranched surfaces in the 3-sphere which satisfy some homological conditions to be embedded in the 3-sphere.

math.GT↗
arXiv2012

Networking Seifert Surgeries on Knots IV: Seiferters and branched coverings

A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pairs of seiferters". One problem of the network is whether there is a path from each vertex to a vertex on a torus knot, the most basic Seifert surgery. We give a method to find seiferters and annular pairs of seiferters for Seifert surgeries obtained by taking two--fold branched covers of tangles. Concerning three infinite families of Seifert surgeries obtained by the second author via branched covers, we find explicit paths in the network from such surgeries to Seifert surgeries on torus knots.

math.GT↗
arXiv2011

Seifert fibered surgeries with distinct primitive/Seifert positions

We call a pair (K, m) of a knot K in the 3-sphere S^3 and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K, m), K can be embedded in a genus 2 Heegaard surface of S^3 in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view. In particular, we can choose such Seifert surgeries (K, m) so that K is a hyperbolic knot whose complement S^3 - K has an arbitrarily large volume.

math.GT↗
arXiv2009

Incompressible surfaces and (1,2)-knots

We give a description of all (1,2)-knots in S^3 which admit a closed meridionally incompressible surface of genus 2 in their complement. That is, we give several constructions of (1,2)-knots having a meridionally incompressible surface of genus 2, and then show that any such surface for a (1,2)-knot must come from one of the constructions. As an application, we show explicit examples of tunnel number one knots which are not (1,2)-knots.

math.GT↗
arXiv2008

The Hexatangle

We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in the 3-sphere. In particular, we want to determine when we get the 3-sphere by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form $({σ_1}^{2e_1})({σ_2}^{2f_1})(σ_2σ_1σ_2)^{2e}$, where $σ_1$, $σ_2$ are the generators of the 3-braid group and $e_1$, $f_1$, $e$ are integers. We study Dehn surgeries on these links, and determine exactly which ones admit an integral surgery producing the 3-sphere. This is equivalent to determining the surgeries of some type on a certain six component link $L$ that produce $S^3$. The link $L$ is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the Hexatangle. Our problem is equivalent to determine which fillings of the spheres by integral tangles produce the trivial knot, which is what we explicitly solve. This hexatangle is a generalization of the Pentangle, which is studied by Gordon and Luecke.

math.GT↗
arXiv2006

Meridional surfaces and (1,1)-knots

We determine all (1,1)-knots which admit an essential meridional surface, namely, we give a construction which produces (1,1)-knots having essential meridional surfaces, and show that if a (1,1)-knot admits an essential meridional surface then it comes from the given construction.

math.GT↗
arXiv2006

Incompressible surfaces and (1,1)-knots

Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the complement of k, then the surface and the knot can be put in a special position, namely, the surface is the boundary of a regular neighborhood of a toroidal graph, and the knot is level with respect to that graph. As an application we show that for any such M there exist tunnel number one knots which are not (1,1)-knots.

math.GT↗
arXiv2006

Acylindrical surfaces in 3-manifolds and knot complements

We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of rational (or alternating) tangles in a projection of a link (or knot). For each g we find knots with tunnel number 2 and manifolds of Heegaard genus 3 containing acylindrical surfaces of genus g. Finally, we construct 3-bridge knots containing quasi-Fuchsinan surfaces of unbounded genus, and use them to find manifolds of Heegaard genus 2 and homology spheres of Heegaard genus 3 containing infinitely many incompressible surfaces.

math.GT↗
arXiv2001

Essential meridional surfaces for tunnel number one knots

We show that for each pair of positive integers g and n, there are infinitely many tunnel number one knots, whose exteriors contain an essential meridional surface of genus g, and with 2n boundary components. We also show that for each positive integer n, there are tunnel number one knots whose exteriors contain n disjoint, non-parallel, closed incompressible surfaces, each of genus n.

math.GT↗