The -Stabilization of a Heegaard Splitting with Distance at Least 6 is Unstabilized

Let M be a compact orientable 3-manifold with M connected. If V ∪S W is a Heegaard splitting of M with distance at least 6, then the -stabilization of V ∪S W along M is unstabilized. Hence M has at least two unstabilized Heegaard splittings with different genera. The basic tool is a result on disk complex given by Masur and Schleimer.     

The Heegaard Genus of Amalgamated 3-Manifolds

Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.     

High distance Heegaard splittings via fat train tracks

We define fat train tracks and use them to give a combinatorial criterion for the Hempel distance of Heegaard splittings for closed orientable 3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on knots in S³.     

Heegaard spines of 3-manifolds

Summary We describe a procedure to construct a 4-coloured graph representing a closed, connected 3-manifold M starting from a Heegaard diagram of M. As a consequence, we prove that, to each Heegaard diagram of a (closed) 3-manifold M, canonically corresponds a spine (Heegaard spine) of M.     

Finite covers of random 3-manifolds

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0.In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π₁(Σ)→Q. If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic. Mathematics Subject Classification (2000) 57M50, 57N10     

Khovanov homology, open books, and tight contact structures

We define the reduced Khovanov homology of an open book (S,?), and identify a distinguished “contact element” in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,?). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology of an open book which parallels the comultiplication on Heegaard Floer homology defined in Baldwin (2008) [4].     

Topologically minimal surfaces versus self-amalgamated Heegaard surfaces

Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.     

Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Let M ₁ and M ₂ be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism ${\phi : {\partial}M_1 \to {\partial}M_2}$ . We analyze the relationship between the sets of low genus Heegaard splittings of M ₁, M ₂, and M, assuming the map ${\phi}$ is “sufficiently complicated”. This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.     

Hyperbolic spatial graphs in 3-manifolds

For any closed connected orientable 3-manifold M, we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S³ via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M.     

Some sufficient conditions for tunnel numbers of connected sum of two knots not to go down

In this paper, we show the following result: Let K _i be a knot in a closed orientable 3-manifold M _i such that (M _i ,K _i ) is not homeomorphic to (S ₂ ×S ₁, x ₀ ×S ₁), i = 1, 2. Suppose that the Euler Characteristic of any meridional essential surface in each knot complement E(K _i ) is less than the difference of one and twice of the tunnel number of K _i . Then the tunnel number of their connected sum will not go down. If in addition that the distance of any minimal Heegaard splitting of each knot complement is strictly more than 2, then the tunnel number of their connected sum is super additive.     

High distance Heegaard splittings of 3-manifolds

J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson-Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553-573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] proved the existence of arbitrarily high distance Heegaard splittings.In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of Mn is at least n.Moreover, we show that π₁(Mn) surjects onto π₁(Mn−1). Hence, if we assume that M⁰ has nontrivial boundary then it follows that the first Betti number β₁(Mn)>0 for all n?1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n?1 and hyperbolizable 3-manifolds for n?3.     

Critical Heegaard surfaces

In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.     

The self-amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable irreducible 3-manifold, F be an essential non-separating closed surface in M. We denote by η(F) the open regular neighborhood of F. If M−η(F) has a high distance Heegaard splitting, then M has a unique minimal Heegaard splitting up to isotopy.     

On the Heegaard Floer homology of branched double-covers

Let be a link. We study the Heegaard Floer homology of the branched double-cover Σ(L) of S³, branched along L. When L is an alternating link, of its branched double-cover has a particularly simple form, determined entirely by the determinant of the link. For the general case, we derive a spectral sequence whose E² term is a suitable variant of Khovanov's homology for the link L, converging to the Heegaard Floer homology of Σ(L).     

Three-manifolds with Heegaard genus at most two represented by crystallisations with at most 42 vertices

It is known that every closed compact orientable 3-manifold M can be represented by a 4-edge-coloured 4-valent graph called a crystallisation of M. Casali and Grasselli proved that 3-manifolds of Heegaard genus g can be represented by crystallisations with a very simple structure which can be described by a 2(g+1)-tuple of non-negative integers. The sum of first g+1 integers is called complexity of the admissible 2(g+1)-tuple. If c is the complexity then the number of vertices of the associated graph is 2c. In the present paper we describe all prime 3-manifolds of Heegaard genus two described by 6-tuples of complexity at most 21.     

On the Heegaard Floer homology of a surface times a circle

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S¹. We determine HF⁺(Σg×S¹,s;C) completely in the case c₁(s)=0, which for g?3 was previously unknown. We show that in this case HF^∞ is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF⁺(Σg×S¹,s;Z) contains nontrivial 2-torsion whenever g?3 and c₁(s)=0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H₁(Σg×S¹) on HF⁺(Σg×S¹,s) when c₁(s) is nonzero.     

Heegaard Genus Formula for Haken Manifolds

Suppose M is a compact orientable 3-manifold and a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of M – Q with respect to the path metric by M ¹, ...,M ^k . Denote the smallest possible genus of a Heegaard splitting of M, or M ^j respectively, for which ∂M, or ∂M ^j respectively, is contained in one compression body by g(M, ∂M), or g(M ^j , ∂M ^j ) respectively. Denote the maximal number of non-parallel essential annuli that can be simultaneously embedded in M ^j by n _j . Then

A formula on Heegaard genus of self-amalgamated 3-manifolds

Let M be an orientable compact irreducible and ∂-irreducible 3-manifold, and suppose ∂M consists of two boundary components F₁ and F₂ with g(F₁)=g(F₂)>1. Let Mf be the closed orientable 3-manifold obtained by identifying F₁ and F₂ via a homeomorphism f:F₁→F₂. With the assumption that M is small or g(M,F₁)=g(M)+g(F₁), we show that if f is sufficiently complicated, then g(Mf)=g(M,∂M)+1.     

Destabilizing Heegaard splittings of knot exteriors

We give a necessary and sufficient condition for Heegaard splittings of knot exteriors to admit destabilizations. As an application, we show the following: let K₁ and K₂ be a pair of knots which is introduced by Morimoto as an example giving degeneration of tunnel number under connected sum. The Heegaard splitting of the exterior of K₁#K₂ derived from certain minimal unknotting tunnel systems of K₁ and K₂ is stabilized.     

LCG Moves in Crystallizations

From a point of view of crystallizations as extended Heegaard diagrams, we introduce LCG ( linear cut and glue) moves with which we investigate relations between geometric T-transformations and crystallization moves. As applications of LCG moves, we verify various equivalences of Lins' crystallizations r _i ⁿ by means of LCG moves. In particular, we show that the two inequivalent Heegaard splittings of the Brieskorn homology sphere (2,3,7) are equivalent under single stabilization of each Heegaard splitting by explicitly showing LCG moves associated with the stable equivalence.