In this paper,the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ∂-irreducible in terms of distances between some kinds of vertex subsets ...In this paper,the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ∂-irreducible in terms of distances between some kinds of vertex subsets of the curve complex and the arc complex of 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
Suppose V ∪S W is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F1 and F2 are pairwise disjoint homeomorphic essential subsurfaces in ?V. In this paper,we give a suffici...Suppose V ∪S W is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F1 and F2 are pairwise disjoint homeomorphic essential subsurfaces in ?V. In this paper,we give a sufficient condition such that the self-amalgamation of V ∪S W along F1 and F2 is unstabilized and uncritical.展开更多
A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator...A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator subgroup of the normal closures in the link group of themeridians excluding themeridian of the last component with the differential induced by removing the last component.Moreover,the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis.These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.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
Let M be a compact connected 3-submanifold of the 3-sphere S^3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S_1, ···, S_n} ...Let M be a compact connected 3-submanifold of the 3-sphere S^3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S_1, ···, S_n} properly embedded in M, ?S = {?S_1, ···, ?S_n}is a complete curve system on F. We call S a complete surface system for M, and ?S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ?S. As an application of the result,it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup Hn.展开更多
The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra,...The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra, which is an approach similar to Jones' s original construction.展开更多
基金Supported in part by NSFC(12071051)the National Science Foundation of Liaoning Province of China(2020-MS-244)the Fundamental Research Funds for the Central Universities(DUT21LAB302)。
文摘In this paper,we will give a sufficient condition for the self-amalgamation of a handlebody to be strongly irreducible.
基金supported by the National Natural Science Foundation of China(No.12071051)。
文摘In this paper,the authors give some sufficient conditions for an amalgamated 3-manifold along a compact connected surface F with boundary to be ∂-irreducible in terms of distances between some kinds of vertex subsets of the curve complex and the arc complex of 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.
基金supported by National Natural Science Foundation of China(Grant Nos.11601209,11671064,11471151,11401069,11329101 and 11431009)Scientific Research Foundation for Doctors of Liaoning Province(Grant No.201601239)+1 种基金Scientific Research Fund of Liaoning Provincial Education Department(Grant No.L201683660)the Fundamental Research Funds for the Central Universities(Grant No.DUT16LK40)
文摘Suppose V ∪S W is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F1 and F2 are pairwise disjoint homeomorphic essential subsurfaces in ?V. In this paper,we give a sufficient condition such that the self-amalgamation of V ∪S W along F1 and F2 is unstabilized and uncritical.
基金The authors would like to thank Joan Birman and Haynes Miller for their encouragements and helpful suggestions on this project.Fuquan Fang and Fengchun Lei supported in part by a Key Grant(No.11431009)an Overseas-Collaboration Grant(No.11329101)of NSFC of ChinaResearch is supported by the Singapore Ministry of Education research Grant(AcRF Tier 1 WBS No.R-146-000-190-112)and a Grant(No.11329101)of NSFC of China.
文摘A link tower is a sequence of links with the structure given by removing the last components.Given a link tower,we prove that there is a chain complex consisting of(non-abelian)groups given by the symmetric commutator subgroup of the normal closures in the link group of themeridians excluding themeridian of the last component with the differential induced by removing the last component.Moreover,the homology groups of these naturally constructed chain complexes are isomorphic to the homotopy groups of the manifold M under certain hypothesis.These chain complexes have canonical quotient abelian chain complexes in Minor’s homotopy link groups with their homologies detecting certain differences of the homotopy link groups in the towers.
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supported by the National Natural Science Foundation of China(Nos.11329101,11431009,11329101,11471151,11401069)the grant of the Fundamental Research Funds for the Central Universities(No.DUT14LK12)
文摘Let M be a compact connected 3-submanifold of the 3-sphere S^3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S_1, ···, S_n} properly embedded in M, ?S = {?S_1, ···, ?S_n}is a complete curve system on F. We call S a complete surface system for M, and ?S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ?S. As an application of the result,it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup Hn.
基金supported by the National Natural Sciences Foundation of China(Nos.11329101,11431009,11301135,11201314,11302136,A2014210062)the Excellent Young Scientist Fund of Shijiazhuang Tiedao University
文摘The authors study the properties of virtual Temperley-Lieb algebra and show how the f-polynomial of virtual knot can be derived from a representation of the virtual braid group into the virtual Temperley-Lieb algebra, which is an approach similar to Jones' s original construction.
