arXiv subjects
Publications and source records attributed to Jesús Rodríguez-Viorato.
In this paper, we study the computation of optimal discrete Morse functions on stratifolds. In particular, we present an algorithm that efficiently computes such functions for a broad class of them. Moreover, we characterize the conditions under which these functions are perfect.
We study compact orientable essential surfaces in knot exteriors in the 3-sphere. The genus $g$, the number of boundary components $b$, and the boundary slope $p/q$ are fundamental invariants of an essential surface. The \textit{realization problem} asks whether, for a given triple $(g, b, q)$ with $g \ge 0$, $b \ge 1$, and $q \ge 1$, there exists a knot $K \subset S^3$ whose exterior $E(K)$ contains a compact orientable essential surface $F$ of genus $g$ with $b$ boundary components and boundary slope $p/q$ for some $p$. In general, not all combinations of $(g, b, q)$ are realizable. First, we show that if $b$ is odd, then $q$ must be equal to $1$. Our main theorem states that for any given even $b \ge 2$ and $q \ge 1$, there exist a genus $g \ge 0$ and a knot $K$ such that $E(K)$ contains a compact orientable essential surface with these parameters.
We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish between universal links and complement universal links and show that these notions do not coincide in general, by exhibiting infinitely many examples of complement universal links that are not universal. Also, we prove that there is no closed aspherical $3$-manifold, such that every closed, aspherical $3$-manifold is a branched covering over it. Finally, we characterize the closed $3$-manifolds admitting branching coverings from $P^3 \# P^3$, and deduce that there is no closed reducible $3$-manifold, such that every closed reducible $3$-manifold is a branched covering over it.
Topological Data Analysis (TDA) is a modern approach to Data Analysis focusing on the topological features of data; it has been widely studied in recent years and used extensively in Biology, Physics, and many other areas. However, financial markets have been studied slightly through TDA. Here we present a quick review of some recent applications of TDA on financial markets, including applications in the early detection of turbulence periods in financial markets and how TDA can help to get new insights while investing. Also, we propose a new turbulence index based on persistent homology -- the fundamental tool for TDA -- that seems to capture critical transitions in financial data; we tested our index with different financial time series (S&P500, Russel 2000, S&P/BMV IPC and Nikkei 225) and crash events (Black Monday crash, dot-com crash, 2007-08 crash and COVID-19 crash). Furthermore, we include an introduction to persistent homology so the reader can understand this paper without knowing TDA.
This paper continues a program due to Motegi regarding universal bounds for the number of non-isotopic essential $n$-punctured tori in the complement of a hyperbolic knot in $S^3$. For $n=1$, Valdez-Sánchez showed that there are at most five non-isotopic Seifert tori in the exterior of a hyperbolic knot. In this paper, we address the case $n=2$. We show that there are at most six non-isotopic, nested, essential 2-holed tori in the complement of every hyperbolic knot.
We prove that there is a knot $K$ transverse to $ξ_{std}$, the tight contact structure of $S^3$, such that every contact 3-manifold $(M, ξ)$ can be obtained as a contact covering branched along $K$. By contact covering we mean a map $φ: M \to S^3$ branched along $K$ such that $ξ$ is contact isotopic to the lifting of $ξ_{std}$ under $φ$.
We give a Python program that is capable to compute and print all the distinct trivalent 2-stratifold graphs up to $N$ white vertices with trivial fundamental group. Our algorithm uses the three basic operations to construct new graphs from any set of given graphs. We iterate this process to construct all the desired graphs. The algorithm includes an optimization that reduces the repetition of generated graphs, this is done by recognizing equivalent white vertices of a graph under automorphism. We use a variation of the AHU algorithm to identify those vertices and as well to distinguish isomorphic graphs in linear time. The returned string from the AHU algorithm is also used as a hashing function to search for repeated graphs in amortized constant time.
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.
Let $t_{α,β}\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $α$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{α,β}$ if $α\geq3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{α,β}$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.
Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating Montesinos knots with three tangles have property $\mathbb{Z}$. We also show that all the pretzel knots of the form $P(p,q,r)$ (not necessarily alternating) have property $\mathbb{Z}$.