arXiv subjects
Publications and source records attributed to Aldo-Hilario Cruz-Cota.
Let $K$ be a polygonal knot in general position with vertex set $V$. A \emph{generic quadrisecant} of $K$ is a line that is disjoint from the set $V$ and intersects $K$ in exactly four distinct points. We give an upper bound for the number of generic quadrisecants of a polygonal knot $K$ in general position. This upper bound is in terms of the number of edges of $K$.
Let $p$ be a branched covering of a Riemann surface to the Riemann sphere $\mathbb{P}^1$, with branching set $B \subset \mathbb{P}^1$. We define the complexity of $p$ as infinity, if $\mathbb{P}^1 \setminus B$ does not admit a hyperbolic structure, or the product of its degree and the hyperbolic area of $\mathbb{P}^1 \setminus B$, otherwise. The topological complexity of a surface $S$ is defined as the infimum of the set of all complexities of branched coverings $M \to \mathbb{P}^1$, where $M$ is a Riemann surface homeomorphic to $S$. We prove that if $S$ is a connected, closed, orientable surface of genus $g$, then its topological complexity, $C_{\text{top}}(S)$, is given by: \[C_{\text{top}}(S)= \left\{ \begin{array}{cl} 2π(2g+1) & \mbox{if } g \geq 1, 6 π& \mbox{if } g=0. \end{array} \right.\]
\noindent Given a Riemann surface $M$, the \emph{complexity} of a branched cover of $M$ to the Riemann sphere $S^2$, of degree $d$ and with branching set of cardinality $n \geq 3$, is defined as $d$ times the hyperbolic area of the complement of its branching set in $S^2$. A branched cover $p \colon M \to S^2$ of degree $d$ is \emph{simple} if the cardinality of the pre-image $p^{-1}(y)$ is at least $d-1$ for all $y \in S^2$. The \emph{(simple) complexity} of $M$ is defined as the infimum of the complexities of all (simple) branched covers of $M$ to $S^2$. We prove that if $M$ is a closed, connected, orientable Riemann surface of genus $g \geq 1$, then: (1) its simple complexity equals $8πg$, and (2) its complexity equals $2π(m_{\text{min}}+2g-2)$, where $m_{\text{min}}$ is the minimum total length of a branch datum realizable by a branched cover $p \colon M \to S^2$.
A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe the Voronoi regions of hex spheres and classify the Voronoi graphs of hex spheres (up to graph isomorphism).
A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2π}{3}$ but less than $2π$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere $M$ satisfies the following properties: (1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of $M$; (2) it embeds isometrically in the 3-dimensional Euclidean space as the boundary of a tetrahedron; and (3) there is a simple closed geodesic $γ$ in $M$ such that a fractional Dehn twist along $γ$ converts $M$ to the double of a parallelogram.