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In this paper, we shall prove that for any integer 0$">, 1) a handlebody of genus 2 contains a separating incompressible surface of genus , 2) there exists a closed 3-manifold of Heegaard genus which contains a separating incompressible surface of genus .

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2.
in this paper we prove that for any positive integer n, 1) a handlebody of genus 2contains a separating incompressible surface of genus n, and 2) there exists a closed 3manifold of heegaard genus 2 which contains a separating incompressible surface of genus n.  相似文献
3.
In this paper, we present a catalogue of all genus two 3-manifolds admitting a contracted triangulation with at most 34 simplexes. Then we give a complete classification of the above manifolds. Mathematics Subject Classifications (2000) 57M05, 57N10, 57M15.Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy) and financially supported by M.I.U.R. of Italy (project Strutture geometriche delle varietà reali e complesse).  相似文献
4.
Let M and M′ be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that Mand M′ are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is ‘sufficiently complicated’, the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of Mand M′ and the genus of their common boundary. Here, a homeomorphism is ‘sufficiently complicated’ if it is the composition of a homeomorphism from the boundary ofM to some surface S, followed by a sufficiently high power of a pseudo-Anosov onS, followed by a homeomorphism to the boundary of M′. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.  相似文献
5.
Suppose M is a compact orientable 3-manifold and a properly embedded orientable boundary incompressible essential surface. Denote the completions of the components of MQ with respect to the path metric by M 1, ...,M k . Denote the smallest possible genus of a Heegaard splitting of M, or M j respectively, for which ∂M, or ∂M j respectively, is contained in one compression body by g(M, ∂M), or g(M j , ∂M j ) respectively. Denote the maximal number of non-parallel essential annuli that can be simultaneously embedded in M j by n j . Then
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6.
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.  相似文献
7.
Let M be a compact connected oriented 3-manifold with boundary, Q1, Q2 C 0M be two disjoint homeomorphic subsurfaces of cgM, and h : Q1 → Q2 be an orientation-reversing homeomorphism. Denote by Mh or MQ1=Q2 the 3-manifold obtained from M by gluing Q1 and Q2 together via h. Mh is called a self-amalgamation of M along Q1 and Q2. Suppose Q1 and Q2 lie on the same component F1 of δM1, and F1 - Q1 ∪ Q2 is connected. We give a lower bound to the Heegaard genus of M when M' has a Heegaard splitting with sufficiently high distance.  相似文献
8.
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.  相似文献
9.
We introduce a method to compute the girth of knots, defined by Hernndez and Lin, using the Jones and BrandtLickorishMillettHo polynomial. We determine the girth of all knots up to 10 crossings.  相似文献
10.
设$M_i~(i=1,2)$是一个紧致可定向的三维流形, $F_i$是$M_i$边界上的一个不可压缩曲面, $M=M_{1}\cup_{f}M_{2}$, 其中$f$是$F_1$到$F_2$一个同胚,对于具有特定条件的相粘曲面$F_i$, 如果$M_i$具有一个Heegaard距离至少是$2(g(M_1)+g(M_2))+1$的Heegaard分解,则$g(M)=g(M_1)+g(M_2)$.  相似文献
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