Research On Several Kinds Of Spatial Graphs And A Kind Of Basic Surface In Their Complements

Low-dimensional topology is an important branch of topology,while knot and link theory is also a significant part in low-dimensional topology.The properties and classifications of incompressible and pairwise incompressible surfaces in knot and link complements have always been important problems in the research of knot and link theory.Based on the results from knot and link theory,experts and scholars in related field at home and abroad have extended the conception of knot and link to spatial graph and studied the properties and classifications of incompressible and pairwise incompressible surfaces in some spatial graph complements.From the definition of n-composite spatial graph,it utilizes the research skills and methods of three-dimensional manifold combinatorial topology theory to construct some specific(2k+3)-composite spatial graph.It also combines the topological graph theory proposed by W.Menasco and C.Adams of incompressible and pairwise incompressible surfaces in knot,link,and spatial graph complements to discuss the properties of incompressible and pairwise incompressible surfaces in several specific kinds of spatial graph complements.Specifically,according to the definition of n-composite spatial graph,it firstly gives three kinds of connection types of integer tangles,and shows that the tangles constructed by those three connection types can be isotoped onto a sphere in S3 while fixing the endpoints of the tangles.Then it constructs some spatial graphs by adding edges to the endpoints of the above tangles,and uses the analytical methods of three-dimensional manifold combinatorial topology to find the minimum number of boundary branches of composite spheres of those spatial graphs in order to prove that all those spatial graphs are(2k+3)-composite spatial graphs.Finally,it chooses three kinds of alternating spatial graphs from those(2k+3)-composite spatial graphs.According to the characteristics of the three kinds of spatial graphs and the topological graph theory proposed by W.Menasco and C.Adams of incompressible and pairwise incompressible surfaces in knot,link,and spatial graph complements,it discusses all the possible combinations of the curves in the topological graphs corresponding to the connected incompressible and pairwise incompressible surfaces in those three kinds of spatial graph complements,which proves that the incompressible and pairwise incompressible surfaces in corresponding alternating spatial graph complements are punctured spheres and the word representations of the corresponding curves in the topological graph of the punctured spheres are Pi(i≥2).