The Properties Of Incompressible Surfaces In A Kind Of Manifolds

3-manifold theory is an important branch of low-dimensional topology.A complicated geometric object can be cut into simple parts along some surfaces properly embedded in3-mafolds.It's an important method to study the topological properties and geometric structures of 3-manifolds.The existence of incompressible surfaces in 3-manifolds is an important research topic for 3-manifold theory.And it will be convenient for us to learn the structure and properties of3-manifolds by studying the incompressible surfaces.In this paper,we can construct a kind of manifolds from the complement of a non-trivial knot by attaching a solid torus along a homeomorphism.Furthermore,the article gives a sufficient condition for the existence of closed incompressible surface in this kind of manifolds.Suppose k is a non-trivial knot,S is a Seifert surface of knot k with minimal genus,andE?k?is the complement of knot k inS³.S?40??E?k?is a closed curve,denoted by l.S?40??E?k?is an orientable surface with a boundary,denoted by P,and?P is l.Suppose T is a solid torus.Let h is a homeomorphism from?E?k?to?T satisfyingh?l?)82(?D₀.Let M?28?E?k??_hTand we can get a new manifold M.P?_l0Dis an orientable closed surface,denoted byP'.The orientable closed surfaceP'is the incompressible surface with minimal genus in M and the manifold M is irreducible,if the conditions are satisfied that?1?There is no essential surfaceS₀inE?k?,andS₀also satisfies the conditions:g?S₀??27?g?S?,?S₀?2,Each boundary component ofS₀is isotopic to l.?2?There is no essential closed surface inE?k?whose genus is less thang?S?.