Heegaard Splittings Research Articles

Volumes of random 3‐manifolds

We prove a law of large numbers for the volumes of families of random hyperbolic mapping tori and Heegaard splittings providing a sharp answer to a conjecture of Dunfield and Thurston.

Orbifolds having Euler number zero Heegaard decomposition

In this paper, we completely classify, up to homeomorphism, the orientable and nonorientable orbifolds which have a Heegaard decomposition consisting of orbifold handlebodies with Euler number zero. In addition we compute their fundamental groups.

Goeritz Groups of Bridge Decompositions

Abstract For a bridge decomposition of a link in the $3$-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the $3$-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. We then give an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the $3$-sphere and the real projective space.

Nielsen equivalence and trisections

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every $$k \ge 2$$ , we exhibit infinitely many manifolds with $$2^{k}-1$$ non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier’s spun trisections. The technique used to distinguish the trisections parallels a common setup for distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes we obtain diffeomorphism invariants. Once the invariance is established, we analyze the Nielsen classes of the spun trisections in terms of the Nielsen classes of the original Heegaard splitting, and leverage work of Lustig, Moriah, and Rosenberger on Nielsen classes of Fuchsian groups in order to distinguish the trisections.

Holonomy perturbations of the Chern-Simons functional for lens spaces

We describe a scheme for constructing generating sets for Kronheimer and Mrowka's singular instanton knot homology for the case of knots in lens spaces. The scheme involves Heegaard-splitting a lens space containing a knot into two solid tori. One solid torus contains a portion of the knot consisting of an unknotted arc, as well as holonomy perturbations of the Chern-Simons functional used to define the homology theory. The other solid torus contains the remainder of the knot. The Heegaard splitting yields a pair of Lagrangians in the traceless $SU(2)$-character variety of the twice-punctured torus, and the intersection points of these Lagrangians comprise the generating set that we seek. We illustrate the scheme by constructing generating sets for several example knots. Our scheme is a direct generalization of a scheme introduced by Hedden, Herald, and Kirk for describing generating sets for knots in $S^3$ in terms of Lagrangian intersections in the traceless $SU(2)$-character variety for the 2-sphere with four punctures.

Heegaard diagrams and optimal Morse flows on non-orientable 3-manifolds of genus 1 and genus $2$


 The present paper investigates Heegaard diagrams of non-orientable closed $3$-manifolds, i.e. a non-orienable closed surface together with two sets of meridian disks of both handlebodies.
 It is found all possible non-orientable genus $2$ Heegaard diagrams of complexity less than $6$.
 Topological properties of Morse flows on closed smooth non-orientable $3$-manifolds are described.
 Normalized Heegaard diagrams are furhter used for classification Morse flows with a minimal number of singular points and singular trajectories
 
 

3D topological models and Heegaard splitting. II. Pontryagin duality and observables

In Paper I [F. Thuillier, “3D topological models and Heegaard splitting I: Partition function,” J. Math. Phys. 60, 32 (2019)], a construction of the smooth Deligne–Beilinson cohomology groups HDp(M) on a closed 3-manifold M represented by a Heegaard splitting XL ∪fXR was presented. Then, the partition functions of the U(1) Chern–Simons and BF Quantum field theories were determined from this construction. In this second and concluding article, we stay in the context of a Heegaard spitting of M to define Deligne–Beilinson 1-currents whose equivalent classes form the elements of HD1(M)⋆, the Pontryagin dual of HD1(M). Finally, we use singular fields to first recover the partition functions of the U(1) Chern–Simons and BF quantum field theories and next to determine the link invariants defined by these theories. The difference between the use of smooth and singular fields is also discussed.

Properties of Casson–Gordon’s rectangle condition

For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

An equivalent description of the lens space L(p,q) with p prime

In the paper, we give an equivalent description of the lens space [Formula: see text] with [Formula: see text] prime in terms of any corresponding Heegaard diagrams as follows: Let [Formula: see text] be a closed orientable 3-manifold with [Formula: see text] and [Formula: see text] a Heegaard splitting of genus [Formula: see text] for [Formula: see text] with an associated Heegaard diagram [Formula: see text]. Assume [Formula: see text] is a prime integer. Then [Formula: see text] is homeomorphic to the lens space [Formula: see text] if and only if there exists an embedding [Formula: see text] such that [Formula: see text] bounds a complete system of surfaces for [Formula: see text].

A sheaf-theoretic model for SL(2,$\mathbb C$) Floer homology

Given a Heegaard splitting of a three-manifold Y , we consider the SL(2, \mathbb C ) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y , i.e., it is independent of the Heegaard splitting. The hypercohomology of the perverse sheaf can be viewed as a model for (the dual of) SL(2, \mathbb C ) instanton Floer homology. We also present a framed version of this construction, which takes into account reducible representations.We give explicit computations for lens spaces and Brieskorn spheres, and discuss the connection to the Kapustin–Witten equations and Khovanov homology.

Kuperberg and Turaev–Viro invariants in unimodular categories

We give a categorical setting in which Penrose graphical calculus naturally extends to graphs drawn on the boundary of a handlebody. We use it to introduce invariants of 3-manifolds presented by Heegaard splittings. We recover Kuperberg invariants when the category comes from an involutory Hopf algebra and Turaev-Viro invariants when the category is semi-simple and spherical.

On tunnel numbers of a cable knot and its companion

Let K be a nontrivial knot in S3 and t(K) its tunnel number. For any (p≥2,q)-slope in the torus boundary of a closed regular neighborhood of K in S3, denoted by K⋆, it is a nontrivial cable knot in S3. Though t(K⋆)≤t(K)+1, Example 1.1 in Section 1 shows that in some case, t(K⋆)≤t(K). So it is interesting to know when t(K⋆)=t(K)+1.After using some combinatorial techniques, we prove that (1) for any nontrivial cable knot K⋆ and its companion K, t(K⋆)≥t(K); (2) if either K admits a high distance Heegaard splitting or p/q is far away from a fixed subset in the Farey graph, then t(K⋆)=t(K)+1. Using the second conclusion, we construct a satellite knot and its companion so that the difference between their tunnel numbers is arbitrary large.

Overcommuting pairs in groups and 3‐manifolds bounding them

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifold and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL_2, related to the Steinberg relation.

Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds

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 finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + \sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.

From Heegaard splittings to trisections; porting 3-dimensional ideas to dimension 4

These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan “Trisections are to 4–manifolds as Heegaard splittings are to 3–manifolds”, focusing on and clarifying the distinction between three ways of thinking of things: the basic definitions as decompositions of manifolds, the Morse theoretic perspective and descriptions in terms of diagrams. We also lay out these themes in two important relative settings: 4–manifolds with boundary and 4–manifolds with embedded 2–dimensional submanifolds.

Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings

By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible but distance at most 3 Heegaard splitting, when its mapping class group is finite. In [19], Qiu and Zou introduced the definition of a locally large distance 2 Heegaard splitting. Extending their definition into a locally large strongly irreducible Heegaard splitting, we proved that its mapping class group is finite. Moreover, for a toroidal 3-manifold which admits a locally large distance 2 Heegaard splitting in [19], we prove that its mapping class group is finite.

On Two Types of Heegaard Diagram Used in Knot Floer Homology

We give an explicit sequence of Heegaard moves interpolating between local versions of the Kauffman-states Heegaard diagram and the planar Heegaard diagram used in knot Floer homology, and show how these local moves can be used to translate between the global versions of the Heegaard diagrams.

Trisections, intersection forms and the Torelli group

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of $S^3$ by regluing according to a map in the kernel of this homomorphism. We prove an analogous result for trisections of 4-manifolds. Specifically, if $X$ and $Y$ admit handle decompositions without 1- or 3-handles and have isomorphic intersection forms, then a trisection of $Y$ can be obtained from a trisection of $X$ by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of homology 3-spheres can be applied, via this result, to obstruct intersection forms of smooth 4-manifolds. As an application, we use the Casson invariant to recover Rohlin's Theorem on the signature of spin 4-manifolds.

Minimal partitions for critical Heegaard splittings

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of [Formula: see text], which is known to be critical, admits a minimal partition. Moreover, we give an example of a critical surface that admits both a minimal partition and a non-minimal partition.

The knot Floer cube of resolutions and the composition product

We examine the relationship between the oriented cube of resolutions for knot Floer homology and HOMFLY-PT homology. By using a filtration induced by additional basepoints on the Heegaard diagram for a knot [Formula: see text], we see that the filtered complex decomposes as a direct sum of HOMFLY-PT complexes of various subdiagrams. Applying Jaeger’s composition product formula for knot polynomials, we deduce that the graded Euler characteristic of this direct sum is the HOMFLY-PT polynomial of [Formula: see text].