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）.