Narrow Results

The intersecting kernel of a Heegaard splitting $H_1{\bigcup }_S{H}_2$ for a compact orientable 3-manifold $M$ is the subgroup $K=\mathrm{\text{Ker}}({i_1}_{\ast })\cap \mathrm{\text{Ker}}({i_2}_{\ast })$ of ${\pi }_1(S)$, where ${i_j}_{\ast }:{\pi }_1(S)\to {\pi }_1(H_j)$ is the homomorphism induced by the inclusion $i_j:S\hookrightarrow H_j$, $j=1,2$. In the paper, we obtain some invariants of 3-manifolds $M$ from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.

The global behavior of nonlinear systems is extremely important in control and systems theory since the usual local theories will only provide information about a system in some neighborhood of an operating point. Away from that point, the system may have totally different behavior and so the theory developed for the local system will be useless for the global one.

In this paper we shall consider the analytical and topological structure of systems on two- and three-manifolds and show that it is possible to obtain systems with "arbitrarily strange" behavior, i.e. arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.

We show that there exist infinitely many 3-manifolds M, each of which admits a genus 4 Heegaard splitting V∪_PW such that there is a Scharlemann-Thompson untelescoping of V∪_PW which decomposes M into three pieces, and that any Casson-Gordon untelescoping of V∪_PW decomposes M into exactly two pieces.

There are exactly four mutually non-isotopic unknotting tunnels τ_i, i = 1,2,3,4 for the pretzel knot P(-2,3,7). Moreover, there are at most 3 non-stabilized genus 3 Heegaard splittings.

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over ℤ[A^±1] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the ℤ₂-homology of the manifold, we determine that they are linearly independent.

Suppose K # K′ is a composite tunnel number two knot where both K and K′ are tunnel number one knots. Let t₁, t₂ be unknotting tunnels of K and t′₁ , t′₂ be unknotting tunnels of K′ and assume that cl(S³ - N(K ∪ t₁ ∪ t₂)) and cl(S³ - N(K′ ∪t′₁ ∪t′₂)) are genus three handlebodies.

Suppose {t₁ , t′₁} , {t₂ , t′₂} are tunnel systems of K # K′. Then we give some condition such that the two genus three Heegaard splittings induced by these tunnel systems become isotopic by one stabilization.

On the other hand, we construct infinitely many examples such that cl(S³ - N(K # K′ ∪ t₁ ∪ t′₁ ∪ t′₂)) is not a genus four handlebody and give candidates of two splittings which cannot be made isotopic by a single stabilization.

Let P, Q be Heegaard surfaces of a closed orientable 3-manifold. In this paper, we introduce a method for giving an upper bound of (Hempel) distance of P by using the Reeb graph derived from a certain horizontal arc in the ambient space [0, 1] × [0, 1] of the Rubinstein–Scharlemann graphic derived from P and Q. This is a refinement of a part of Johnson's arguments used for determining stable genera required for flipping high distance Heegaard splittings.

In this paper, we give infinitely many non-Haken hyperbolic genus three 3-manifolds each of which has a finite cover whose induced Heegaard surface from some genus three Heegaard surface of the base manifold is reducible but can be compressed into an incompressible surface. This result supplements [A. Casson and C. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987) 275–283] and extends [J. Masters, W. Menasco and X. Zhang, Heegaard splittings and virtually Haken Dehn filling, New York J. Math. 10 (2004) 133–150].

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. We find examples of elements of the mapping class group that are periodic, reducible and pseudo-Anosov on the Heegaard surface, but are isotopy trivial in the ambient manifold. We prove structural theorems about the first two classes, in particular showing that if a periodic element is trivial in the mapping class group of the ambient manifold, then the manifold is not hyperbolic.

In this paper, we prove that a tunnel number two knot induces a critical Heegaard splitting in its exterior if there are two weak reducing pairs such that each weak reducing pair contains the cocore disk of each tunnel. Moreover, we prove that a connected sum of two 2-bridge knots or more generally that of two (1,1)-knots always contains a critical Heegaard splitting in its exterior as the examples of the main theorem.

For a boundary-reducible 3-manifold M with ∂M a genus-g surface, we show that if M admits a genus-(g + 1) Heegaard surface S, then the disk complex of S is simply connected. Also we consider the connectedness of the complex of reducing spheres. We investigate the intersection of two reducing spheres for a genus-3 Heegaard splitting of (torus) × I.

Let $M$ be a $3$-manifold, $F$ be an essential planar surface which cuts $M$ into two 3-manifolds $M_1$ and $M_2$. Suppose $M_i=V_i{\bigcup }_{S_i}{W}_i$$(i=1,2)$ is a Heegaard splitting of $M_i$, $M=V{\bigcup }_{S}W$ is the dual Heegaard splitting of $M_1=V_1{\bigcup }_{S_1}{W}_1$ and $M_2=V_2{\bigcup }_{S_2}{W}_2$ along $F$, where $V=V_1\cup (D\times [-1,2])\cup V_2$ and $W=cl(M-V)$. In this paper, we give a condition of unstabilized dual Heegaard splittings of $3$-manifolds by using Hempel’s distance and the method of proof of Gordon’s Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of $3$-manifolds for any Hempel’s distance.

A $2n$-component non-splittable spatial graph $G=G_1\cup \cdots \cup G_{2n}$ in $S^3$ possibly admits a smooth involution on its exterior whose fixed point set is a closed surface $F$ of genus $g\ge 1$. It is known that $n\ge g$ holds if $G$ is a graph link in $S^3$. In this paper, we consider $G$ to be a hyperbolic spatial graph in a closed orientable 3-manifold $M$. We first study the case where $M$ is the 3-sphere, and provide a method for constructing $G$ and $F$ with $a_1+\cdots +a_n>g>1$, where $1-a_i$ is the Euler characteristic of $G_i$ for $1\le i\le n$. Next, we study the case where $M$ is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries.

Let $M_i=V_i{\cup }_{S_i}{W}_i$$(i=1,2)$ be a Heegaard splitting of compact orientable $3$-manifold, and $M=V{\cup }_{S}W=(V_1{\cup }_{S_1}{W}_1){\cup }_F(V_2{\cup }_{S_2}{W}_2)$ be an amalgamation along an essential surface $F$. In this paper, we show that if there is an essential disk $D_2$ in $V_2$, such that $D_2$ cuts $V_2$ into two product $I$-bundles, one of which is $F\times [0,1]$, and $d(S_i)\ge 3$$(i=1,2)$, then $M=V{\cup }_{S}W$ is unstabilized and $S$ is uncritical. We also give a sufficient condition for $S$ to be critical.

In the paper, we give an equivalent description of the lens space $L(p,q)$ with $p$ prime in terms of any corresponding Heegaard diagrams as follows: Let $M$ be a closed orientable 3-manifold with ${\pi }_1(M)\ne 1,$ and $U{\cup }_{F}V$ a Heegaard splitting of genus $n$ for $M$ with an associated Heegaard diagram $(U;J_1,\dots ,J_n)$. Assume $p$ is a prime integer. Then $M$ is homeomorphic to the lens space $L(p,q)$ if and only if there exists an embedding $f:U\hookrightarrow L(p,q)$ such that $L=\{f(J_1),\dots ,f(J_n)\}$ bounds a complete system of surfaces for $W=\overline{L(p,q)\backslash f(U)}$.

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of $(\mathrm{\text{a closed orientable surface}})\times S^1$, 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.

In this paper, we consider decompositions of 3-manifolds with three handlebodies. We classify such decompositions of the 3-sphere and lens spaces with small genera. These decompositions admit operations called stabilizations. We also determine whether these decompositions are obtained by stabilization from another decomposition.

In this paper we study isotopy classes of closed connected orientable surfaces in the standard $3$-sphere. Such a surface splits the $3$-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a $2$-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial $1$-handles can recover the original surface. Therefore, we can reduce the study of surfaces in the $3$-sphere to that of $2$-component handlebody-links up to stabilizations. Then, by using $G$-families of quandles, we construct invariants of $2$-component handlebody-links up to attaching trivial $1$-handles, which lead to invariants of surfaces in the $3$-sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the $3$-sphere.