Each Thurston geometry admits a Heegaard genus two example

An algorithm to determine the Heegaard genus of a 3-manifold

We give an algorithmic proof of the theorem that a closed orientable irreducible and atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. The proof gives an algorithm to determine the Heegaard genus of an atoroidal 3-manifold.

Manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings

We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed.

Heegaard genus and complexity of fibered knots

We prove that if a fibered knot $K$ with genus greater than one in a three-manifold $M$ has a sufficiently complicated monodromy, then $K$ induces a minimal genus Heegaard splitting $P$ that is unique up to isotopy, and small genus Heegaard splittings of $M$ are stabilizations of $P$. We provide a complexity bound in terms of the Heegaard genus of $M$. We also provide global complexity bounds for fibered knots in the three-sphere and lens spaces.

Degree one maps between small 3-manifolds and Heegaard genus

We prove a rigidity theorem for degree one maps between small 3-manifolds using Heegaard genus, and provide some applications and connections to Heegaard genus and Dehn surgery problems.

Surface bundles with genus two Heegaard splittings

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with genus two Heegaard splittings and show that each is the result of integral Dehn surgery in one of these Seifert fibered spaces along loops where the Heegaard surface intersects a horizontal surface. (This type of surgery preserves both the bundle structure and the Heegaard splittin…

Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting

An updated proof of a 1933 theorem of Goeritz, exhibiting a finite set of generators for the group of automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. The group is analyzed via its action on a certain connected 2-complex. (The analogous problem for higher genus Heegaard splittings appears to remain unresolved.)

Computing Heegaard genus is NP-hard

We show that {\sc Heegaard Genus $\leq g$}, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to $g$, is NP-hard. The result follows from a quadratic time reduction of the NP-complete problem {\sc CNF-SAT} to {\sc Heegaard Genus $\leq g$}.

Quantum invariants can provide sharp Heegaard genus bounds

Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group.

3-Manifolds with irreducible Heegaard splittings of high genus

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. Many of the manifolds have irreducible Heegaard splittings of arbitrary large genus. All these splittings are horizontal and are isotopic, after one stabilization, to a multiple stabilization of certain canonical low genus vertical Heegaard splittings.

Invariant Heegaard Surfaces in Manifolds with Involutions and the Heegaard Genus of Double Covers

Let M be a 3-manifold admitting a strongly irreducible Heegaard surface S and f:M \to M an involution. We construct an invariant Heegaard surface for M of genus at most 8 g(S) - 7. As a consequence, given a (possibly branched) double cover π:M \to N we obtain the following bound on the Heegaard genus of N: g(N) \leq 4g(S) - 3. We also get a bound on the complexity of the branch set in terms of g(S). If we assume that M is non-Haken, by Casson and Gordon we may replace g(S) by g(M) in all the statements above.

On unstabilzed genus three critical Heegaard surfaces

Let $M$ be a compact orientable irreducible 3-manifold and $H$ be an unstabilized genus three Heegaard splitting of $M$. In this article, we will define a simplicial complex of weak reducing pairs for $H$ and find several properties of this complex. Using this method, we will prove that an unstabilized Heegaard splitting of genus three is critical in a certain condition. In addition, we will show that the standard genus three Heegaard splitting for $T^3$ and the induced Heegaard splitting of the three component chain exterior by a certain tunnel system are critical as examples of the main theo…

Berge's distance 3 pairs of genus 2 Heegaard splittings

Following an example discovered by John Berge, we show that there is a 4-component link L \subset (S^1 x S^2)#(S^1 x S^2) so that, generically, the result of Dehn surgery on L is a 3-manifold with two inequivalent genus 2 Heegaard splittings, and each of these Heegaard splittings is of Hempel distance 3.

可约三维流形的等变Heegaard亏格

Equivariant Heegaard genus of reducible 3-manifolds

本文研究了有限群作用下可约3-流形的等变Heegaard亏格，证明其在等变连通和下具有超加性、加性或次加性，并利用三维轨形薄位置理论给出了该亏格的精确界。

On the disk complexes of weakly reducible, unstabilized Heegaard splittings of genus three III - Generalized Heegaard splittings and mapping classes

Let $M$ be an orientable, irreducible $3$-manifold admitting a weakly reducible genus three Heegaard splitting as a minimal genus Heegaard splitting. In this article, we prove that if $[f]$, $[g]\in Mod(M)$ give the same correspondence between two isotopy classes of generalized Heegaard splittings consisting of two Heegaard splittings of genus two, say $[\mathbf{H}]\to[\mathbf{H}']$, then there exists a representative $h$ of the difference $[h]=[g]\cdot[f]^{-1}$ such that (i) $h$ preserves a suitably chosen embedding of the Heegaard surface $F'$ obtained by amalgamation from $\mathbf{H}'$ whic…

The asymptotic behaviour of Heegaard genus

Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let {M_i} be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong Heegaard genus of M_i cannot grow more slowly than the square root of d_i. (3) If the Heegaard genus of M_i grows more slowly than the fourth root of d_i, then M_i fibres over the circle for all sufficiently large i. These results provide supporting evidence for the Heegaard gradie…

Automorphisms of the three-torus preserving a genus three Heegaard splitting

The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. For the genus three Heegaard splitting of the 3-torus, we find an eight element generating set for this group. Six of these generators induce generating elements of the mapping class group of the 3-torus and the remaining two are isotopy trivial in the 3-torus.

Heegaard genus formula for Haken manifolds

Given a 3-manifold M containing an incompressible surface Q, we obtain an inequality relating the Heegaard genus of M and the Heegaard genera of the components of M - Q. Here the sum of the genera of the components of M - Q is bounded above by a linear expression in terms of the genus of M, the Euler characteristic of Q and the number of parallelism classes of essential annuli for which representatives can be simultaneously imbedded in the components of M - Q.

Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two

Let ${\mathcal H}$ be the group of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve a Heegaard splitting of genus two. In this paper, we use a tree in the barycentric subdivision of the disk complex of a handlebody of the splitting to obtain a finite presentation of ${\mathcal H}$.

Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots II

A knot K is called a 1-genus 1-bridge knot in a 3-manifold M if (M,K) has a Heegaard splitting (V_1,t_1)\cup (V_2,t_2) where V_i is a solid torus and t_i is a boundary parallel arc properly embedded in V_i. If the exterior of a knot has a genus 2 Heegaard splitting, we say that the knot has an unknotting tunnel. Naturally the exterior of a 1-genus 1-bridge knot K allows a genus 2 Heegaard splitting, i.e., K has an unknotting tunnel. But, in general, there are unknotting tunnels which are not derived form this procedure. Some of them may be levelled with the torus \partial V_1=\partial V_2, who…

Reducing spheres of genus-2 Heegaard splitting of $S^3$

The Goeritz group of the standard genus-g Heegaard splitting of the three sphere, $G_g$, acts on the space of isotopy classes of reducing spheres for this Heegaard splitting. Scharlemann MR2199366 (2007c:57020) uses this action to prove that $G_2$ is finitely generated. In this article, we give an algorithm to construct any reducing sphere from a standard reducing sphere for a genus-2 Heegaard splitting of the $S^3$. Using this we give an alternate proof of the finite generation of $G_2$ assuming the finite generation of the stabilizer of the standard reducing sphere.

Thinning genus two Heegaard spines in the 3-sphere

We study trivalent graphs in $S^{3}$ whose closed complement is a genus two handlebody. We show that such a graph, when put in thin position, has a simple (i. e. non-loop) level edge.

Hyperbolic volume, Heegaard genus and ranks of groups

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.

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 M and 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 M and 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 of M to some surface S, followed by a sufficiently high power of a pseudo-Anosov on S, fol…

Small knots of large Heegaard genus

Building off ideas developed by Agol, we construct a family of hyperbolic knots $K_n$ whose complements contain no closed incompressible surfaces and have Heegaard genus exactly $n$. These are the first known examples of small knots having large Heegaard genus. Using work of Futer and Purcell, we are able to bound the crossing number for each $K_n$ in terms of $n$.

The mapping class groups of reducible Heegaard splittings of genus two

The manifold which admits a genus-$2$ reducible Heegaard splitting is one of the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their connected sums. For each of those manifolds except most lens spaces, the mapping class group of the genus-$2$ splitting was shown to be finitely presented. In this work, we study the remaining generic lens spaces, and show that the mapping class group of the genus-$2$ Heegaard splitting is finitely presented for any lens space by giving its explicit presentation. As an application, we show that the fundamental groups of the spaces of the genus-$…

An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary

We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein's work on almost normal surfaces, and also a new structure called a partially flat angled ideal triangulation.

Degeneration of Heegaard genus, a survey

We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing together the boundary components of surface times I, and (3) gluing a handlebody to the boundary of a 3-manifold. We detail those cases in which it is known when the Heegaard genus is less than what is expected after gluing.

A topologically minimal, weakly reducible, unstabilized Heegaard splitting of genus three is critical

Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological index of F is two if F is topologically minimal.

Genus two Heegaard splittings of exteriors of 1-genus 1-bridge knots

A knot K in a closed connected orientable 3-manifold M is called a 1-genus 1-bridge knot if (M,K) has a splitting into two pairs of a solid torus V_i (i=1,2) and a boundary parallel arc in it. The splitting induces a genus two Heegaard splitting of the exterior of K naturally, i.e., K has an unknotting tunnel. However the converse is not true in general. Then we study such general case in this paper. One of the conclusions is that the unknotting tunnel may be levelled with the torus \partial V_1=\partial V_2.

具有 Heegaard 亏格二的 3-流形的紧叶状结构

Taut foliations of 3-manifolds with Heegaard genus two

本文证明了具有 Heegaard 亏格二的闭可定向不可约 3-流形，若其基本群左可序，则必存在可余定向的紧叶状结构，揭示了群序性质与叶状结构之间的联系。

Complexity and Heegaard genus of an infinite class of compact 3-manifolds

Using the theory of hyperbolic manifolds with totally geodesic boundary, we provide for every integer n greater than 1 a class of such manifolds all having Matveev complexity equal to n and Heegaard genus equal to n+1. All the elements of this class have a single boundary component of genus n, and the numbers of distinct members of the class grows at least exponentially with n.

New Heegaard Floer slice genus and clasp number bounds

We define several concordance invariants using knot Floer homology which give improvements over known slice genus and clasp number bounds from Heegaard Floer homology. We also prove that the involutive correction terms of Hendricks and Manolescu give both a slice genus and a clasp number bound.

Non-uniqueness of high distance Heegaard splittings

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $Σ$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this, proving that in such a manifold, every other Heegaard surface for $M$ of genus $g' < \frac{d}{2}$ is a stabilization of $Σ$. In the present paper, we show that Hartshorn's bound is sharp and Scharlemann-Tomova's bound is very close to sharp. In particular, for every pair of integers $g \geq 2, d \geq 2$, we construct a three-manifold $M$ with a genus $g$, distance $d…

A universal bound for surfaces in 3-manifolds with a given Heegaard genus

It is shown that for given positive integers g and b, there is a number C(g,b), such that any orientable compact irreducible 3-manifold of Heegaard genus g has at most C(g,b) disjoint, nonparallel incompressible surfaces with first Betti number b_1 < b.

The Heegaard genus of bundles over S^1

This paper explores connections between Heegaard genus, minimal surfaces, and pseudo-Anosov monodromies. Fixing a pseudo-Anosov map phi and an integer n, let M_n be the 3-manifold fibered over S^1 with monodromy phi^n. JH Rubinstein showed that for a large enough n every minimal surface of genus at most h in M_n is homotopic into a fiber; as a consequence Rubinstein concludes that every Heegaard surface of genus at most h for M_n is standard, that is, obtained by tubing together two fibers. We prove this result and also discuss related results of Lackenby and Souto.

Genus two Heegaard splittings of orientable three-manifolds

It was shown by Bonahon-Otal and Hodgson-Rubinstein that any two genus-one Heegaard splittings of the same 3-manifold (typically a lens space) are isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert manifolds have distinct genus-two Heegaard splittings. In an earlier paper, we presented a technique for comparing Heegaard splittings of the same manifold and, using this technique, derived the uniqueness theorem for lens space splittings as a simple corollary. Here we use a similar technique to examine, in general, ways in which two non-isotopic genus-…

Multiple genus 2 Heegaard splittings: a missed case

A gap in a paper of Rubinstein-Scharlemann is explored: new examples are found of closed orientable 3-manifolds with possibly multiple genus 2 Heegaard splittings. Properties common to all the examples in the original paper are not universally shared by the new examples: some of the new examples have Hempel distance 3, and it is not clear that a single stabilization always makes the multiple splittings isotopic.

Heegaard genus of the connected sum of m-small knots

We prove that if $K_1 \subset M_1,...,K_n \subset M_n$ are m-small knots in closed orientable 3-manifolds then the Heegaard genus of $E(#_{i=1}^n K_i)$ is strictly less than the sum of the Heegaard genera of the $E(K_i)$ ($i=1,...,n$) if and only if there exists a proper subset $I$ of $\{1,...,n\}$ so that $#_{i \in I} K_i$ admits a primitive meridian. This generalizes the main result of Morimoto in \cite{morimoto1}.

The Goeritz group of a Heegaard splitting of genus two of a Seifert manifold whose base orbifold is sphere with three exceptional points of sufficiently complex coefficients

In this paper, we add examples to Goeritz groups, the mapping class groups of given Heegaard splittings of 3-manifolds. We focus on a Heegaard splitting of genus two of a Seifert manifold whose base orbifold is sphere with three exceptional points of sufficiently complex coefficients, where "sufficiently complex" means that every surgery coefficient p_{l} over q_{l} of each exceptional fiber (in a surgery description) satisfies that q_{l} is congruent to neither +1 or -1 modulo p_{l}.

Obtaining genus 2 Heegaard splittings from Dehn surgery

Let K' be a hyperbolic knot in S^3 and suppose that some Dehn surgery on K' with distance at least 3 from the meridian yields a 3-manifold M of Heegaard genus 2. We show that if M does not contain an embedded Dyck's surface (the closed non-orientable surface of Euler characteristic -1), then the knot dual to the surgery is either 0-bridge or 1-bridge with respect to a genus 2 Heegaard splitting of M. In the case M does contain an embedded Dyck's surface, we obtain similar results. As a corollary, if M does not contain an incompressible genus 2 surface, then the tunnel number of K' is at most 2…

An upper bound on common stabilizations of Heegaard splittings

We show that for any two Heegaard splittings of genus $p$ and $q$ for the same closed 3-manifold, there is a common stabilization of genus at most 3/2 p + 2q - 1. One may compare this to recent examples of Heegaard splittings whose smallest common stabilizations have genus at least $p+q$ or $p + 1/2 q$ depending on the notion of equivalence.

Bounding the stable genera of Heegaard splittings from below

We describe for each postive integer $k$ a 3-manifold with Heegaard surfaces of genus $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We also show that for every positive $n$, there is a 3-manifold that has $n$ pairwise non-isotopic Heegaard splittings of the same genus all of which are stabilized.

Heegaard Splittings of Twisted Torus Knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain "twisted torus knots" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. W…

Heegaard diagrams and Floer homology

We review the construction of Heegaard Floer homology for closed three-manifolds and also for knots and links in the three-sphere. We also discuss three applications of this invariant to knot theory: studying the Thurston norm of a link complement, the slice genus of a knot, and the unknotting number of a knot. We emphasize the application to the Thurston norm, and illustrate the theory in the case of the Conway link.

Large embedded balls and Heegaard genus in negative curvature

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g >= 1/2 cosh(r) where r denotes the radius of any isometrically embedded ball in M. Assuming an unpublished result of Pitts and Rubinstein improves this to g >= 1/2 cosh(r) + 1/2. We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.

Stabilizations of Heegaard splittings of graph manifolds

We show that after one stabilization, a strongly irreducible Heegaard splitting of suitably large genus of a graph manifold is isotopic to an amalgamation along a modified version of the system of canonical tori in the JSJ decomposition. As a corollary, two strongly irreducible Heegaard splittings of a graph manifold of suitably large genus are isotopic after at most one stabilization of the higher genus splitting.

$3$维球面亏格-$3$ Heegaard分裂的Powell猜想

The Powell Conjecture for the genus-three Heegaard splitting of the $3$-sphere

本文通过证明亏格-$3$约化球复形的连通性，证实了$3$维球面亏格-$3$ Heegaard分裂的Powell猜想，并给出向高亏格推广的框架。

Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds

Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus two Heegaard splittings for non-prime 3-manifolds, which is a generalization of the result of Montesinos-Safont. Finally, we show that the mapping class group of the splitting, called the Goeritz group, is finitely …

Heegaard Floer correction terms and rational genus bounds

Given an element in the first homology of a rational homology 3-sphere $Y$, one can consider the minimal rational genus of all knots in this homology class. This defines a function $Θ$ on $H_1(Y;\mathbb Z)$, which was introduced by Turaev as an analogue of Thurston norm. We will give a lower bound for this function using the correction terms in Heegaard Floer homology. As a corollary, we show that Floer simple knots in L-spaces are genus minimizers in their homology classes, hence answer questions of Turaev and Rasmussen about genus minimizers in lens spaces.

Bridge number, Heegaard genus and non-integral Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.

Handlebody-preserving finite group actions on Haken manifolds with Heegaard genus two

Let $M$ be a closed orientable 3-manifold with a genus two Heegaard splitting $(V_1, V_2; F)$ and a non-trivial JSJ-decomposition, where all components of the intersection of the JSJ-tori and $V_i$ are not $\partial$-parallel in $V_i$ for $i=1,2$. If $G$ is a finite group of orientation-preserving diffeomorphisms acting on $M$ which preserves each handlebody of the Heegaard splitting and each piece of the JSJ-decomposition of $M$, then $G\cong \mathbb{Z}_2$ or $\mathbb{D}_2$ if $V_j\cap(\cup T_i)$ consists of at most two disks or at most two annuli.

A Presentation For The Automorphisms Of The 3-Sphere That Preserve A Genus Two Heegaard Splitting

Scharlemann constructed a connected simplicial 2-complex $Γ$ with an action by the group ${\mathcal H_{2}}$ of isotopy classes of orientation preserving homeomorphisms of $S^3$ that preserve the isotopy class of an unknotted genus 2 handlebody $V$. In this paper we prove that the 2-complex $Γ$ is contractible. Therefore we get a finite presentation of ${\mathcal H_{2}}$.

Knot exteriors with additive Heegaard genus and Morimoto's Conjecture

Given integers g_i > 1 (i=1,...,n) we prove that there exist infinitely may knots K_i in S^3 so that g(E(K_i)) = g_i and the Heegaard genus of the exterior of the connected sum of K_1,...,K_n is the sum the Heegaard genera of K_1,...,K_n, that is: g(E(K_1#...#K_n)) = g(E(K_1)) +...+ g(E(K_n)). (Here, E() denotes the exterior and g() the Heegaard genus.) Together with Theorem 1.5 of [1], this proves the existence of counterexamples to Morimoto's Conjecture (Conjecture 1.5 of [2]). [1] Tsuyoshi Kobayashi and Yo'av Rieck. On the growth rate of the tunnel number of knots. J. Reine Angew. Math., 59…

Alternate Heegaard genus bounds distance

Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized copy of P or the Hempel distance of the splitting P is no greater than twice the genus of Q. More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either a) P and Q can be well-separated or b) P and Q are isotopic or c) the Hempel distance of P is no greater than twice the genus of Q.