Let Hn be an orientable handlebody of genus n. It has been proved that for n not less than 2, there exists an annulus-busting curve in δHm. In the present paper, we prove that for n not less than 2, there exists an e...Let Hn be an orientable handlebody of genus n. It has been proved that for n not less than 2, there exists an annulus-busting curve in δHm. In the present paper, we prove that for n not less than 2, there exists an essential simple closed curve C in OHm which intersects each essential planar surface in Hn non-emptily. Furthermore, we show that for n not less than 3, a pants-busting curve must also be an annulus-busting curve.展开更多
For a handlebody H w ith dH = 5, let F belong S be an essential connected subsurface of S. Let C（S） be the curve complex of S, AC（F） be the arc and curve complex of F , D（H） belong C（S） be ...For a handlebody H w ith dH = 5, let F belong S be an essential connected subsurface of S. Let C（S） be the curve complex of S, AC（F） be the arc and curve complex of F , D（H） belong C（S） be the disk complex of H and πf（D （H ） ） belong AC（F） be the image of D （H） in AC （F）. We introduce the definition of subsurface 1-distance between the 1-simplices of AC（F） and show that under some hypothesis, πF（D（H） ） comes within subsurface 1-distance at most 4 of every 1-simplex of AC （F）.展开更多
The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1)The annulus sum H=H1∪AH2 of two han...The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1)The annulus sum H=H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2;(2)Let H=H1∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S=Sg,b,b 1,ni=g(Hi)2,i=1,2.Suppose that S is incompressible in both H1 and H2.Then H is a handlebody if and only if there exists a basis J={J1,...,Jm}with a partition(J1,J2)of J such that J1 is primitive in H1 and J2 is primitive in H2.展开更多
Let H be a handlebody, J={J 1, …, J n} a collection of 2 sided pairwise disjoint simple closed curves on H. The 3 manifold obtained from H by attaching 2 handles to H along the curves in J is called an n re...Let H be a handlebody, J={J 1, …, J n} a collection of 2 sided pairwise disjoint simple closed curves on H. The 3 manifold obtained from H by attaching 2 handles to H along the curves in J is called an n relator 3 manifold, and is denoted as H J. In this paper, a sufficient and necessary condition for H J to be a handlebody is described.展开更多
Applying the Morse theory,we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies.We also give a necessary and sufficient conditio...Applying the Morse theory,we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies.We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form.Our algorithm is practical.Several examples are given to test the algorithm.展开更多
In the present paper, we characterize the bordered 3-manifolds with genus 1 D-splittings and SD-splittings. We also describe a minimal genus SD-splitting for a handlebody and surface × I.
基金The grant (09XBKQ09) of Harbin Normal Universitythe NSF (11101058) of ChinaChina Postdoctoral Science Foundation (2011M500049)
文摘Let Hn be an orientable handlebody of genus n. It has been proved that for n not less than 2, there exists an annulus-busting curve in δHm. In the present paper, we prove that for n not less than 2, there exists an essential simple closed curve C in OHm which intersects each essential planar surface in Hn non-emptily. Furthermore, we show that for n not less than 3, a pants-busting curve must also be an annulus-busting curve.
文摘For a handlebody H w ith dH = 5, let F belong S be an essential connected subsurface of S. Let C（S） be the curve complex of S, AC（F） be the arc and curve complex of F , D（H） belong C（S） be the disk complex of H and πf（D （H ） ） belong AC（F） be the image of D （H） in AC （F）. We introduce the definition of subsurface 1-distance between the 1-simplices of AC（F） and show that under some hypothesis, πF（D（H） ） comes within subsurface 1-distance at most 4 of every 1-simplex of AC （F）.
基金supported by National Natural Science Foundation of China(Grant Nos.11431009 and 11671064)the Fundamental Research Funds for the Central Universities(Grant No.DUT19LK15)Ministry of Science and Education of Russia(Grant No.1.13557.2019/13.1)。
文摘The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1)The annulus sum H=H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2;(2)Let H=H1∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S=Sg,b,b 1,ni=g(Hi)2,i=1,2.Suppose that S is incompressible in both H1 and H2.Then H is a handlebody if and only if there exists a basis J={J1,...,Jm}with a partition(J1,J2)of J such that J1 is primitive in H1 and J2 is primitive in H2.
文摘Let H be a handlebody, J={J 1, …, J n} a collection of 2 sided pairwise disjoint simple closed curves on H. The 3 manifold obtained from H by attaching 2 handles to H along the curves in J is called an n relator 3 manifold, and is denoted as H J. In this paper, a sufficient and necessary condition for H J to be a handlebody is described.
基金supported by National Natural Science Foundation of China(Grant No.12071051)the Fundamental Research Funds for the Central Universities(Grant No.DUT21LAB302).
文摘Applying the Morse theory,we give a standard form for a class of surfaces which includes all the properly embedded incompressible surfaces in 3-dimensional handlebodies.We also give a necessary and sufficient condition to determine the incompressibility of such surfaces placed in our standard form.Our algorithm is practical.Several examples are given to test the algorithm.
基金Work supported in part by a grant (No.10171024) of NSFCa grant of the Outstanding Youth Fellowship of Heilongjiang Province
文摘In the present paper, we characterize the bordered 3-manifolds with genus 1 D-splittings and SD-splittings. We also describe a minimal genus SD-splitting for a handlebody and surface × I.
