Related papers: Filling with separating curves

A filling pair $(\alpha, \beta)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $\alpha\cup\beta$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be…

Let $S_{g}$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_{g}$ and intersect minimally, by showing that such orbits are in…

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of…

A pair $(\alpha, \beta)$ of simple closed curves on a surface $S_{g,n}$ of genus $g$ and with $n$ punctures is called a filling pair if the complement of the union of the curves is a disjoint union of topological disks and once punctured…

Let $S_g$ denote the genus $g$ closed orientable surface. A \emph{coherent filling pair} of simple closed curves, $(\alpha,\beta)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its…

Let $F_g$ be a closed orientable surface of genus $g$. A set $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if…

A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected. The length of a…

Let $ S_g $ be a closed surface of genus $ g $ and let $ (\alpha, \beta) $ be a filling pair on $ S_g $; then $ i(\alpha, \beta) \geq 2g-1 $, where $ i $ is the (geometric) intersection number. Aougab and Huang demonstrated that…

Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $\gamma_i$ and $\gamma_j$ are in minimal position, is…

Let $S_g$ denoting the genus $g$ closed orientable surface. An {\em origami} (or flat structure) on $S_g$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom…

A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of…

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of collections of $2g+1$ simple closed curves on $S_g$ which pairwise intersect exactly once, extending a result of the…

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for…

We show that the asymptotic growth rate for the minimal cardinality of a set of simple closed curves on a closed surface of genus $g$ which fill and pairwise intersect at most $K\ge 1$ times is $2\sqrt{g}/\sqrt{K}$ as $g \to \infty$ . We…

Let $S_{g}$ denote the closed orientable surface of genus $g$. In joint work with Huang, the first author constructed exponentially-many (in $g$) mapping class group orbits of pairs of simple closed curves whose complement is a single…

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $\beta:S \to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,\beta)$…

An "origami" (or flat structure) on a closed oriented surface, $S_g$, of genus $g \geq 2$ is obtained from a finite collection of unit Euclidean squares by gluing each right edge to a left one and each top edge to a bottom one. The main…

Let $S_g$ denote a closed oriented surface of genus $g \geq 2$. A set $\Omega = \{ c_1, \dots, c_d\}$ of pairwise non-homotopic simple closed curves on $S_g$ is called a filling system or simply a filling of $S_g$, if $S_g\setminus \Omega$…

Let $\Sigma$ be a smooth projective surface, let $f' : S' \to \Sigma$ be a double cover of $\Sigma$ and let $\mu : S \to S'$ be the canonical resolution. Put $f = f'\circ\mu$. An irreducible curve $C$ on $\Sigma$ is said to be a splitting…