相关论文: Haken spheres for genus two Heegaard splittings

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

It is well-known that Heegaard genus is additive under connected sum of 3-manifolds. We show that Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the…

Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a…

We describe an example of a closed orientable 3-manifold with distinct distance three genus two Heegaard splittings. This demonstrates that the constructions of alternate genus two Heegaard splittings of closed orientable 3-manifolds…

Representing Z/N as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/N, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/N to construct an associated…

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a…

We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the…

A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the complex of curves on a surface and a Heegaard splitting as a pair of subcomplexes generated by the equivalent diagrams. We relate geometric and combinatorial…

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give…

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly…

The Heegaard genus g of an irreducible closed orientable 3-manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if p of…

This is a "software upgrade" to a paper originally published in 1976, with cleaner statements and improved proofs. The main result is that, in a Haken 3-manifold, the space of all incompressible surfaces in a single isotopy class is…

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be…

We propose a way to derive polynomial invariants of closed, orientable $3$-manifolds from Heegaard diagrams via cellularly embedded graphs. Given a Heegaard diagram of an irreducible $3$-manifold $M$, we associate a Heegaard graph $G\subset…

Heegaard splittings provide a natural representation of closed 3-manifolds by gluing two handlebodies along a common surface. These splittings can be equivalently given by two finite sets of meridians lying on the surface, which define a…

In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound…

Hedden defined two knots in each lens space that, through analogies with their knot Floer homology and doubly pointed Heegaard diagrams of genus one, may be viewed as generalizations of the two trefoils in S^3. Rasmussen shows that when the…

Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The…

The $k$-cut complex was recently introduced by Bayer et al. as a generalization of earlier work of Fr{\"o}berg (1990) and Eagon and Reiner (1998), and was shown to be shellable for several classes of graphs. In this article, we prove that…

The rectangle condition for a genus $g$ Heegaard splitting of a 3-manifold, defined by Casson and Gordon, provides a sufficient criterion for the Heegaard splitting to be strongly irreducible. However it is unknown whether there exists a…