Some Properties Of Knots With (g,2)-decompositions And 3-manifolds Containing Essential Annuli

Studying the topological and geometric properties of 3-manifolds by surfaces contained in 3-manifolds is an important method of the theory of 3-manifolds.In this paper,we study some properties of 3-manifolds with torus boundary containing separating essential annuli and knots with(g,2)-decompositions.When gluing two general 3-manifolds along an surface in their boundaries,some new properties will be generated,when gluing two 3-manifolds with torus boundaries along along an annulus in their boundaries,there is a new property under some special condition,we will give it in subsection 3.1.Finally,for knots with(g,2)-decompositions,we give a general result in subsection 3.2, i.e,let K(?)S³ be a knot,then g_n(K)≤g_n-1(K)+1(n≥2).In addition,we give some results under some special condition.