On(e)-reducible 3-manifolds Which Admit Complete Surface Systems

In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M , then there exists a complete surface system S for M , which is equivalent to S′, such that D is disjoint from S . We also obtain some properties of such 3-manifolds which can be embedded in S3.     

Three-bridge links with infinitely many three-bridge spheres

We construct infinitely many three-bridge links each of which admits infinitely many three-bridge spheres up to isotopy.     

组合群论中的一个定理

Let F= F(X) be a free group of rand n, A be a finite subset of F(X) and x∈X be a generator. The theorem states that x can be denoted as a rotation-inserting word of A if x is in the normal closure of A in F(X). Finally, an application of the theorem in Heegaard splitting of 3manifolds is given.     

Normal surfaces as combinatorial slicings

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. Focus is given to the case of dimension 3, where slicings are (discrete) normal surfaces. For the cases of 2-neighborly 3-manifolds as well as quadrangulated slicings, lower bounds on the number of quadrilaterals of slicings depending on its genus g are presented. These are shown to be sharp for infinitely many values of g. Furthermore, we classify slicings of combinatorial 3-manifolds which are weakly neighborly polyhedral maps.     

字定理在Heegaard分解方面的一个应用

The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus of Heegaard splitting of 3-manifolds is given. This leads Poincare conjecture to a new formulation.     

A note on tunnel number of composite knots

Let K be a knot in a sphere S³. We denote by t(K) the tunnel number of K. For two knots K₁ and K₂, we denote by K₁?K₂ the connected sum of K₁ and K₂. In this paper, we will prove that if one of K₁ and K₂ has high distance while the other has distance at least 3 then t(K₁?K₂)=t(K₁)+t(K₂)+1.     

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.     

Local detection of strongly irreducible Heegaard splittings

Let S be a Heegaard splitting surface of a compact orientable 3-manifold M. If S is strongly irreducible, the manner in which it can intersect a ball or a solid torus in M is very constrained and the allowable configurations are simple and useful. Splitting surfaces not conforming to these simple local pictures must be weakly reducible.     

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].     

A Sufficient Condition for the Genus of an Annulus Sum of Two 3-manifolds to Be Non-degenerate

Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2.