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Register a new userFilling of closed Surfaces
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Authors
Bidyut Sanki
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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 fillings of $F_g$. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of $F_g$ are in the same $text{Mod}(F_g)$-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of $F_2$ whose complement is a single disc (i.e., a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of $text{Mod}(F_2)$. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of $F_2$ is two. Finally, given positive integers $g$ and $k$ with $(g, k) eq (2, 1)$, we construct a filling pair of $F_g$ such that the complement is a union of $k$ topological discs.
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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 $F_gsetminus Omega$ is a union of $b$ topological discs for some $bgeq 1$. A filling system is called emph{minimal}, if $b=1$. The emph{size} of a filling is defined as the number of its elements. We prove that the maximum size of a filling of $F_g$ with $b$ complementary discs is $2g+b-1$. Next, we show that for $ggeq 2, bgeq 1text{ with }(g,b) eq (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2leq sleq 2g+b-1$ (resp. $3leq sleq 2g+b-1$), there exists a filling of $F_g$ of size $s$ with $b$ complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For $ggeq 2$, we show that for a minimal filling $Omega$ of size $s$, the emph{geometric intersection numbers} satisfy $max leftlbrace i(gamma_i, gamma_j)| i eq jrightrbraceleq 2g-s+1$, and for each such $s$ there exists a minimal filling $Omega=leftlbrace gamma_1, dots, gamma_s rightrbrace$ such that $maxleftlbrace i(gamma_i, gamma_j) | i eq jrightrbrace = 2g-s+1$.
This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $varepsilon$-fills the surface.
We introduce the concept of a bridge trisection of a neatly embedded surface in a compact four-manifold, generalizing previous work with Alexander Zupan in the setting of closed surfaces in closed four-manifolds. Our main result states that any neatly embedded surface $mathcal{F}$ in a compact four-manifold $X$ can be isotoped to lie in bridge trisected position with respect to any trisection $mathbb{T}$ of $X$. A bridge trisection of $mathcal{F}$ induces a braiding of the link $partialmathcal{F}$ with respect to the open-book decomposition of $partial X$ induced by $mathbb{T}$, and we show that the bridge trisection of $mathcal{F}$ can be assumed to induce any such braiding. We work in the general setting in which $partial X$ may be disconnected, and we describe how to encode bridge trisected surface diagrammatically using shadow diagrams. We use shadow diagrams to show how bridge trisected surfaces can be glued along portions of their boundary, and we explain how the data of the braiding of the boundary link can be recovered from a shadow diagram. Throughout, numerous examples and illustrations are given. We give a set of moves that we conjecture suffice to relate any two shadow diagrams corresponding to a given surface. We devote extra attention to the setting of surfaces in $B^4$, where we give an independent proof of the existence of bridge trisections and develop a second diagrammatic approach using tri-plane diagrams. We characterize bridge trisections of ribbon surfaces in terms of their complexity parameters. The process of passing between bridge trisections and band presentations for surfaces in $B^4$ is addressed in detail and presented with many examples.
The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations of surfaces. This paper proves that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $geq1$, when the approximating triangle meshes are reasonably good. To the best of the authors knowledge, this is the first convergence result on computing uniformizations for surfaces of genus $>1$.
We present a practical algorithm to test whether a 3-manifold given by a triangulation or an ideal triangulation contains a closed essential surface. This property has important theoretical and algorithmic consequences. As a testament to its practicality, we run the algorithm over a comprehensive body of closed 3-manifolds and knot exteriors, yielding results that were not previously known. The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. Our methods are relevant for other difficult computational problems in 3-manifold theory, such as the recognition problem for knots, links and 3-manifolds.
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