The IPI Surfaces In Alternating Link And Spatial Graph Complements

The properties and classification of incompressible surfaces are not only the core content of3-manifolds but also the important tool to classify 3-manifolds.Since the eighties of last century,some experts and scholars at home or abroad who do research on 3-manifolds theory have done a lot of researches about the properties and classification of incompressible pairwise incompressible surfaces in link and spatial graph complements,and they have obtained a series of significant and useful results.Which include that W.Menasco proved that in alternating link complements,if the number of perforation is less than six,the incompressible pairwise incompressible surface must be punctured sphere,in [1],and he has also proved the isotopic limited of incompressible pairwise incompressible surfaces which peforation's number is finite,In [2] W.Menasco proved that if the topological graph corresponding the surfaces in alternating link complements satisfied specific properties,we can distinguish whether the surface corresponding toplogical graph is incompressible pairwise incompressible surface or not.In[3],professor Han You Fa proved that in almost alternating link complement,the incompressible pairwise incompressible surfce must be punctured sphere if the number of perforation of the surface is less than eight.In[4],Colin.Adams proved the incompressible pairwise incompressible surface must be punctured sphere in alternating almost alternating space,if the number of perforation of the space is less than seven.In[5],professor Han You Fa proved that the incompressible pairwise incompressible surface in alternating link complement must be punctured sphere,if the twist crossing number of the pretzed link is less than six.In[6],Shuxin Wang,Xue Yan proved that incompressible pairwise incompressible surface in a class of alternating spatial graph complements must be punctured sphere.This paper start with the angle of combination topology of 3-manifolds,by using the skills and methords of combination topology of 3-manifolds,in addition,the results about incompressible pairwise incompressible surface in alternating link and spatial graph complements.This paper constructs a link satisfing there exist incompressible pairwise incompressible surface which has arbitary genus but less than n in a certain complementary space for arbitary positive integer n.This paper generalized a series results that obtained by W.Menasco and Han You Fa,find a series of alternating links,at the same time,give the incompressible pairwise incompressible surface with big genus in link complements.The results of this paper have positive and directed action on study the properties and the classification of link and spatial graph complements.Particularly,the results of this paper canapply to study the classification of link by useing linear decomposition of link and the additivity of genus of bounded surfaces of 3-manifolds.