Related papers: The rectangle condition does not detect the strong…

In this paper, we give an isotopy classification of 3-bridge spheres of 3-bridge arborescent links, which are not Montesinos links. To this end, we prove a certain refinement of a theorem of J.S. Birman and H.M. Hilden on the relation…

We prove a structure theorem for 3-manifolds with non-trivial JSJ-decomposition and 2-generated fundamental group. We deduce a variety of Corollaries. Note this is not a complete classification of such manifolds. In particular we believe…

We show that a smooth embedding of a closed 3-manifold in S^3 x R can be isotoped so that every generic level divides S^3 x t into two handlebodies (i.e., is Heegaard) provided the original embedding has a unique local maximum with respect…

We consider 3-manifolds given as Heegaard splittings $M=H^-\cup_\Sigma H^+$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular,…

Using basic properties of one-sided Heegaard splittings, a direct proof that geometrically compressible one-sided splittings of RP^3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that two-sided…

Given a Heegaard surface in the $3$-sphere, we show that any non-separating weak reducing pair for the surface admits a reducing sphere that separates the two disks of the pair if and only if the genus of the surface is at most $3$.

We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken…

We show that if $M$ is a fibered, orientable 3-manifold, and if $\pi_1 M$ has 1-relator presentation, then the presentation is induced by a Heegaard splitting of $M$. A corollary is that, for these manifolds, the rank of $\pi_1 M$ is equal…

Let $T$ be a separating incompressible torus in a 3-manifold $M$. Assuming that a genus $g$ Heegaard splitting $V \cup_S W$ can be positioned nicely with respect to $T$ (e.g. $V \cup_S W$ is strongly irreducible), we obtain an upper bound…

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is…

Let $g \ge 2$ and assume that we are given a genus $g$ Heegaard splitting of a closed orientable $3$-manifold with the distance greater than $2g+2$. We prove that the mapping class group of the once-stabilization of such a Heegaard…

We show that every p-fold strictly-cyclic branched covering of a b-bridge link in the 3-sphere admits a p-symmetric Heegaard splitting of genus g=(b-1)(p-1). This gives a complete converse to a result of Birman and Hilden, and gives an…

Let M_1 and M_2 be closed, orientable 3-manifolds. Let H_i denote a Heegaard surface in M_i. We prove that if H_1 # H_2 comes from stabilizing a lower genus splitting of M_1 # M_2 then either H_1 or H_2 comes from stabilizing a lower genus…

Let $Y_1$ be a closed, oriented 3-manifold and $\Sigma$ denote a non-separating closed, orientable surface in $Y_1$ which consists of two connected components of the same genus. By cutting $Y_1$ along $\Sigma$ and re-gluing it using an…

This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation…

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…

We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any…

We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegard genus, and the cut number. We show that j_$ is an invariant of weak p-congruence.

We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose…

An open book decomposition of a 3-manifold $M$ induces a Heegaard splitting for $M$, and the minimal genus among all Heegaard splittings induced by open book decompositions is called the \emph{open book genus} of $M$. It is conjectured by…