Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find

左可序性与可定向紧致叶面

Left orderability and taut foliations with orderable cataclysm

本文研究了3维流形的左可序性，通过可定向紧致叶面和伪Anosov流的性质，证明了特定条件下的流形基群的左可序性，并探讨了Dehn填充的应用。

Stable equivalence of handlebody decompositions whose partitions are multibranched surfaces

In this paper, we consider decompositions of closed orientable 3-manifolds with more than 3 handlebodies, where the union of intersections of handlebodies is a multibranched surface. We define stabilization operations for such decompositions and show the stable equivalence.

Computing the crosscap number of a knot using integer programming and normal surfaces

The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varying trade-offs between precision and speed: (i) an algorithm based on Hilbert basis enumeration and (ii) an algorithm based on exact integer programming, both of which either compute the solution precisely or reduce it to two possible values, and (iii) a fast but limited precision…

Compact 3-manifolds via 4-colored graphs

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana's school in Modena. In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumera…

Exhaustion of the curve graph via rigid expansions

For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion, are closely related to Aramayona and Leiniger's finite rigid set, and in fact a consequence of our proof is that Aramayona and Leininger's set also exhausts the curve graph via rigid expansions.

闭曲面上最小相交填充曲线系统的渐近计数

Asymptotic enumeration of minimally intersecting filling curve systems on closed surfaces

本文证明了闭曲面上最小相交填充曲线系统的映射类群轨道数满足渐近公式$N_g \sim 16^g(2g)!/(64\pi\sqrt{2}g^3)$，并研究单曲线情形，改进上界，同时精确计算小亏格值并给出猜想。

A New Decomposition Theorem for 3-Manifolds

Let M be a (possibly non-orientable) compact 3-manifold with (possibly empty) boundary consisting of tori and Klein bottles. Let $X\subset\partial M$ be a trivalent graph such that $\partial M\setminus X$ is a union of one disc for each component of $\partial M$. Building on previous work of Matveev, we define for the pair (M,X) a complexity c(M,X) and show that, when M is closed, irreducible and P^2-irreducible, $c(M,\emptyset)$ is the minimal number of tetrahedra in a triangulation of M. Moreover c is additive under connected sum, and, given any n>=0, there are only finitely many irreducible…

Fox reimbedding and Bing submanifolds

Let M be an orientable closed connected 3-manifold. We introduce the notion of amalgamated Heegaard genus of M with respect to a closed separating 2-manifold F, and use it to show that the following two statements are equivalent: (i) a compact connected 3-manifold Y can be embedded in M so that the exterior of the image of Y is a union of handlebodies; and (ii) a compact connected 3-manifold Y can be embedded in M so that every knot in M can be isotoped to lie within the image of Y . Our result can be regarded as a common generalization of the reimbedding theorem by Fox [Fox48] and the charact…

Deformation of Hyperbolic Cone-Structures: Study of the non-Colapsing case

This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence (M_{i},p_{i}) of pointed hyperbolic cone-manifolds with topological type (M,Σ), where M is a closed, orientable and irreducible 3-manifold and Σ an embedded link in M. Assuming that the lengths of the singularity remain uniformly bounded, we prove that either the sequence M_{i} collapses and M is Seifert fibered or a Sol manifold, or the sequence M_{i} does not collapse and in this case a subsequ…

Detecting surface bundles in finite covers of hyperbolic closed 3-manifolds

The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal surface in the cover $M'$, splitting $M'$ into a disjoint union of $q$ handlebodies and compression bodies. We show that there exists a fiber in the complement of $F$ provided that $d$, $q$ and $g$ satisfy some inequality involving an explicit constant $k$ depending only on the volume and the injectivity radius of $M$. In particular, this theorem applies to…

On Reeb graphs induced from smooth functions on closed or open manifolds

For a smooth function on a smooth manifold of a suitable class, the space of all connected components of preimages is the graph and called the {\it Reeb graph}. Reeb graphs are fundamental tools in the algebraic and differential topological theory of Morse functions and more general functions which are not so wild. In this paper, we study whether we can construct a smooth function with good geometric properties inducing a given graph as the Reeb graph. This problem has been essentially launched by Sharko in 2000s and various answers have been given by Masumoto, Michalak, Saeki, and so on. Rece…