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].

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].

Derivation of Schubert normal forms of 2-bridge knots from (1,1)-diagrams

In this paper, we show that the dual [Formula: see text]-diagram of a [Formula: see text]-diagram (a.k.a. a two-pointed genus one Heegaard diagram) [Formula: see text] with [Formula: see text] and [Formula: see text] is given by [Formula: see text] where [Formula: see text] is the multiplicative inverse of [Formula: see text] modulo [Formula: see text] with [Formula: see text]. We also present explicitly how to derive a Schubert normal form of a [Formula: see text]-bridge knot from the dual [Formula: see text]-diagram of [Formula: see text] using weakly K-reducibility of [Formula: see text]-decompositions. This gives an alternative proof of Grasselli and Mulazzani’s result asserting that [Formula: see text] is a [Formula: see text]-diagram of the [Formula: see text]-bridge knot with a Schubert normal form [Formula: see text].

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.

COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

The Dual and Mirror Images of the Dunwoody 3-Manifolds

Recently, in 2013, we proved that certain presentations present the Dunwoody3-manifold groups. Since the Dunwoody3-manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody3-manifolds. It is well known that every(1,1)-knots in a lens space can be represented by the set𝒟of the 4-tuples(a,b,c,r)(Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody3-manifolds, we proved the fact that the certain subset of𝒟representing all2-bridge knots of(1,1)-knots is determined completely by using the dual and mirror(1,1)-decompositions (S. H. Kim and Y. Kim (2011)). In this paper, we show how to obtain the dual and mirror images of all elements in𝒟as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).

Parity criterion for unstabilized Heegaard splittings

We give a parity condition of a Heegaard diagram implying that it is unstabilized. As applications, we show that Heegaard splittings of 2-fold branched coverings of n-component, n-bridge links in S3 are unstabilized, and we also construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.

On the contact Ozsváth-Szabó invariant

Sarkar and Wang proved that the hat version of Heegaard Floer homology group of a closed oriented 3-manifold is combinatorial starting from an arbitrary nice Heegaard diagram and in fact every closed oriented 3-manifold admits such a Heegaard diagram. Plamenevskaya showed that the contact Ozsváth-Szabó invariant is combinatorial once we are given an open book decomposition compatible with a contact structure. The idea is to combine the algorithm of Sarkar and Wang with the recent description of the contact Ozsváth-Szabó invariant due to Honda, Kazez and Matić. Here we observe that the hat version of the Heegaard Floer homology group and the contact Ozsváth-Szabó invariant in this group can be combinatorially calculated starting from a contact surgery diagram. We give detailed examples pointing out to some shortcuts in the computations.