Naturality and Mapping Class Groups in Heegaard Floer Homology

We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.

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

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].

Isometry Groups of Some Dunwoody Manifolds

Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces (possibly 𝕊3) branched over (1,1)-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.

Heegaard diagrams corresponding to Turaev surfaces

We describe a correspondence between Turaev surfaces of link diagrams on S2 ⊂ S3 and special Heegaard diagrams for S3 adapted to links.

CONSTRUCTING DOUBLY-POINTED HEEGAARD DIAGRAMS COMPATIBLE WITH (1,1) KNOTS

A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parametrization of this family of knots by a four-tuple of integers, which they call Schubert's normal form. This paper presents an algorithm for constructing a genus 1 doubly-pointed Heegaard diagram compatible with K, given a Schubert's normal form for K. The construction, coupled with results of Ozsváth and Szabó, provides a practical way to compute knot Floer homology groups for (1,1) knots. The construction uses train tracks, and its method is inspired by the work of Goda, Matsuda, and Morifuji.